The Pennsylvania State University the Graduate School Eberly College of Science

The Pennsylvania State University The Graduate School Eberly College of Science

ALIE-ALGEBRAICAPPROACHTOTHE

LOCALINDEXTHEOREMON

COMPACTHOMOGENEOUSSPACES

A Dissertation in Mathematics by Seunghun Hong

c 2012 Seunghun Hong

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

August 2012 The dissertation of Seunghun Hong was reviewed and approved∗ by the following:

Nigel Higson Evan Pugh Professor of Mathematics Dissertation Advisor Chair of the Doctoral Committee

John Roe Professor of Mathematics

Ping Xu Professor of Mathematics

Martin Bojowald Associate Professor of Physics

Yuxi Zheng Professor of Mathematics Head of the Department of Mathematics

∗Signatures are on ﬁle in the Graduate School.

ii ABSTRACT

Using a K-theory point of view, R. Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This dissertation explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to N. Berline and M. Vergne. But the use of B. Kostant’s cubic Dirac operator in place of the Rieman- nian Dirac operator leads to substantial simpliﬁcations. An important role is also played by the quantum Weil algebra of A. Alekseev and E. Meinrenken.

iii CONTENTS

LISTOFSYMBOLS vi

ACKNOWLEDGMENTS viii

1 INTRODUCTION 1

2 REVIEWOFTHETHEORYOFCOMPACTLIEGROUPS 10 2.1 Analytical Aspects...... 10 2.2 Algebraic Aspects...... 17 2.3 The Weyl Character Formula and the Spectrum of the Laplacian...... 26

3 THEVOLUMEOFACOMPACTLIEGROUP 39 3.1 The Euler-Maclaurin Formula...... 40 3.2 An Example: SU(2) ...... 42 3.3 The General Case...... 45

4 THEHEATKERNELOFTHELAPLACIANON A COMPACT LIE GROUP 50 4.1 The Duﬂo Isomorphism...... 51 4.2 The Asymptotic Expansion of the Heat Kernel...... 58

5 THEQUANTUMWEILALGEBRA 66 5.1 Clifford Algebras...... 66 5.2 The Quantum Weil Algebra...... 77 5.3 The Classical Weil Algebra...... 82 5.4 The Relative Weil Algebra...... 87

6 EQUIVARIANT DIFFERENTIAL OPERATORS AND THEQUANTUMWEILALGEBRA 92 6.1 Principal Bundles and Associated Vector Bundles...... 92 6.2 Clifford Module Bundles and Spin Manifolds...... 99 6.3 Equivariant Differential Operators and the Relative Weil Algebra...... 104

7 THE ASYMPTOTIC EXPANSION OF THE HEAT KERNEL OF THE CUBIC DIRAC OPERATOR 109 7.1 The Convolution Kernel of a Generalized Laplacian.. 110

iv 7.2 The Asymptotic Expansion of the Heat Kernel of D/ (g, k)2 ...... 112

8 THELOCALINDEXTHEOREMON COMPACT HOMOGENEOUS SPACES 118 8.1 Review of the Heat Kernel Proof of the Local Index Theorem...... 119 8.2 The Local Index Theorem on G/K ...... 124

9 THEDISTRIBUTIONALINDEXOF THE CUBIC DIRAC OPERATOR 137 9.1 The Distributional Index of a Transversally Elliptic Operator...... 138 9.2 The Distributional Index of D/ (g, k) ...... 143

BIBLIOGRAPHY 149

v LISTOFSYMBOLS

A, 134 D/ (g, k), 88 exp∗, 53

Abas, 87 D/ ±, 119 expCl, 74

Ak-bas, 88 D/ g/k, 107 expV , 59 Ag, 87 D(A), 51 f ∼ g, 17 Ak, 87 D(G), 17 G fT , 33 Ahor, 87 D(G) , 58 Fr(F), 93 Ak-hor, 87 D(g), 58 FrO(F), 93 Aˆ , 123 D(g)G, 58 FrSO(F), 93 ad, 11 De, 52 Ad, 11 ∆G, 13 g¯, 77 Adf, 113 ∆g, 58 gC, 29 ads, 71 ∆ˆg, 84 gC,α, 29 Altλ, 34 δG, 53 gˆ, 78 Alt(µ), 142 δg, 53 Gb, 19 δij, 23 γ, 75 [ , ]g, 77 D(g, k), 90 γg, 88 [ , ] s, 69, 71 p diagW, 90 γ , 88 0 c, 100 D (M), 55 Γ(S), 4 Duf, 55 Γ 2E(G) Cµ, 141 , 37 Duf, 58 gr A C(G, C)ﬁn, 24 , 56 C (G, C)ﬁn, E, 125 HP, 94 ∞ 24 e, 10 HpP, 94 ch, 123 eI, 68 ht, 15 χu, 18 e, 69 Cl(M), 99 I ind, 119 E0 (A), 52 l(M), 99 e Ind C 0 K, 140 E0(g), 53 Cl(V), 67 inds, 119 E0 (g)G, 55 Cl(n), 67 0 ι , 72 0 v k Ee(G), 53 Cl (V), 69 ιX, 77, 83 0 G Ee(G) , 55 Clk(V), 68 ε, 52 j(X), 55, 130 Cl(V), 67 εG, 58 jk(X), 130

D/ , 78, 81 εg, 58 jg/k(X), 130

vi ∗ κ, 12 P ×p iE, 93 TKM, 138

2 trV , 134 Lcl(G, C), 25 Q, 84 trg, 59 λ, 75 q, 69 p λ , 90, 125 uˇ, 18 Rf, 140 Λcoroot, 31 ± U(g), 20 Rf , 140 Λe, 28 Uk(g), 56 R(G), 35 Λg, 31 Rb(G), 37 [V], 35 ΛT , 28 ρ, 31 V(µ), 35 LX, 77, 83 ρg, 90 vol, 13 ν, 125 VP, 93 S, 102, 113 ± VpP, 93 Ω, 22 S , 75 Ω b g, 81 Sn, 67 W, 26 Ω (P; E) S(g) bas , 95 , 53 W(g), 82, 97 Ω(M; P × E) Sk(g) ν , 95 , 56 W(g, k), 90 O(tn) σ , 16 , 57 W(g), 77 O(t ) σ , 17 k, 56 W(g, k), 88 ∞ SpanR, 57 P0, 121 Str, 121 X, 77 PBW, 53 supp, 141 Xb, 78 Φ, 31 Xe, 10, 93 Φ+, 30 τ, 20, 51 Xee, 11 π, 125 T(g), 20 Ξ, 127 PSpin(M), 101 ΘV , 133

P ×G E, 92 ΘT , 133 Z(g), 23

vii ACKNOWLEDGMENTS

It is a great pleasure to express my gratitude to the people who have helped me to ﬁnish this dissertation. I am grateful, foremost, to Professor Nigel Higson for introducing me to this beautiful area of mathematics. It is an honor for me to be a student of his. I am also grateful for the generous mental and ﬁnancial support that he has provided. It is through Professor Loring Tu that I learned equivariant cohomology and the Weil algebra, which has helped me tremendously in my research. As a matter of fact, had I not met him while I was a physics student at Tufts, I would not have switched my discipline and this dissertation would not be. I learned a great amount of mathematics (especially geometric functional analysis) from Professor John Roe through his lectures, talks, and books, and also through the various conversations we had. I thank Professor Ping Xu for his several thought-provoking com- ments on my research. Professor Martin Bojowald gave interesting feedbacks from the physics side. I learned the basics of K-theory and K-homology from Professor Paul Baum. To be around him has been joyful; I have always learned something when I was with him. When I had questions, he listened very carefully, and cheerfully shared his insights. Professor Alberto Bressan taught the course on second-order linear partial differential equations when I took it. I am thankful for his taking time in discussing some of my questions surrounding the heat semigroup. I would like to thank Professor Adrian Ocneanu for the vibrant discussions we had on representation theory. As the Korean proverb says, “in a mizzle, one is not aware of his/her clothes getting wet”; so it is impossible for me to fully per- ceive all that I have absorbed from the intellectually stimulating and diverse environment promoted by the faculty of the Department of Mathematics at PENN STATE. But I do wish to mention that I owe much of my education to, in addition to the aforementioned profes- sors, Professor Anatole Katok, Professor Anna Mazzucato, Professor Kris Wysocki, Professor Nate Brown, Professor Steve Simpson, Pro- fessor Winnie Li, and Professor Yuri Zarkhin. I would like to also thank the staff members of the Department of Mathematics at PENN STATE. Especially Ms. Becky Halpenny; she has every reason to be called the “grad student mama.”

viii Many people in the worldwide LATEX community have shared their work and ideas online; thanks to them, I was able to focus more on the content than the typesetting of this dissertation. It is impossible for me to fully appreciate all that I have received from my parents, Samsun Hong and Beomsik Roh; words can not fully express my gratitude. Additional thanks goes to my father for his comment on typography. I would also like to express my heartfelt gratitude to my parents- in-law, Sunggook Kim and Youngsook Jung, for their love and support they have provided in various ways and forms. My greatest debt of love is to my wife Jeongwoon Kim — for her love, care, and patience. I must thank our daughter Zoyoung Hong too, for she has been an unceasing fount of cheer. A personal goal of mine was to complete the dissertation before she asks “When will you ﬁnish, daddy?” I think I made it. Finally, I owe sincere and earnest thankfulness to my church family, the congregation of Grace Presbyterian Church, for their support and prayers. And, of course, to the One who has listened and an- swered their prayers, Christ Jesus, the only wise God.

ix Spectral point of view is the one which appears from experiments, when you study the universe, this is no fantasy. — A. Connes [49]

INTRODUCTION1

ROM the sound waves of a drum to a quantum particle trapped F in a box, many fundamental physical systems can be studied by solving the eigenvalue problem of the Laplacian under Dirichlet boundary conditions:

−∆φ = λφ, on U, (1.0.1) φ = 0, on ∂U, where U is a bounded open subset of a Euclidean space. It is a standard result of elliptic partial differential operator theory that there is an orthonormal basis for L2(U) consisting of solutions to Equa- tion 1.0.1. Moreover, the basis vectors can be ordered in such a way that their corresponding eigenvalues { λk }k=1 form an unbounded sequence of nondecreasing positive real numbers.∞ A signiﬁcant question is “how fast does λk grow?” This simple question is related to one of the most fascinating chapters in the history of science — the problem of blackbody radiation and the birth of quantum physics. Consider a cube of side length L with hollow interior, and walls that completely block any light. The intensity of the electromagnetic waves inside the cube that is under thermodynamic equilibrium was studied in the late 19th century, and it led to the dis- covery of Planck’s constant h. To calculate the intensity, one needs to calculate the total number of standing waves that have frequency less than f; call that number N(f). The standing waves are the solutions of Equation 1.0.1 where U is the interior of the cube. Using the method of separation of variables, one ﬁnds that the eigenvalues are of the form π2 λ = (`2 + m2 + n2), L2 where `, m, n are positive integers. Physically, λ is related to the

1 frequency f of the standing wave by λ = (2πf)2. Thus, N(f) is equal to the number of lattice points in the positive 3 octant of R that are bound by radius R := 2fL. Thus, N(f) can be approximated by 1/8 of the volume of the ball that has radius R. The error of this estimate vanishes as f tends to inﬁnity, and we have 4π N(f) = f3L3 + o(f3). (1.0.2) 3 Because of the thermal origin of the blackbody radiation, this relation is expected1 to hold even when U is an arbitrarily shaped cavity. In other words, we expect the following to hold for any bounded 3 open subset U of R : N(f) 4π lim = vol(U), (1.0.3) f f3 3 where vol(U) is the volume of U. →∞ Indeed, H. Weyl [94, 95] proved the asymptotic law N(λ) vol(U) lim = , (1.0.4) λ n/2 n/2 n λ (4π) Γ( 2 + 1) n where U is a bounded→∞ open subset of R with n = 2 or 3. Equa- tion 1.0.4 is known as Weyl’s law. L. Gårding [46] proved it for higher-dimensions (for generic elliptic operators). For the Laplacian on a closed Riemannian manifold U, the same law was proved by S. Minakshisundaram and Å. Pleijel [75], even for generic elliptic operators by J. J. Duistermaat and V. W. Guillemin [33]. Weyl’s law tells us, in particular, that if we know the spectrum of the Laplacian then from it we can calculate the volume of the ambi- ent space U. In this regard (and for other more) Weyl’s law, as N. Higson puts it [56, p. 456], is the ﬁrst theorem of noncommutative geometry.

HEAT TRACE. Weyl’s law can be reformulated as an asymptotic behavior of the heat trace

Z(t) = tr(et∆) = e−tλk . (1.0.5) ∞ k=1 X It resembles the “partition function” in physics — a function that is often invariant under the symmetry of the physical system it de- scribes. (Let us not worry about the convergence of the sum at this point.) The relation between Z(t) and the number of eigenvalues under certain cutoff becomes evident if we write Z(t) as N Z(t) = lim e−tλk . N k=1 X 1 It was H. A. Lorentz [71] who formally→∞ posed this as a mathematical problem.

2 In the limit t 0+, the partial sum N → e−tλk k=1 X converges to N, which is the number of the eigenvalues from λ1 to λN. Clearly the inﬁnite sum Z(t) tends to inﬁnity as t 0+; but with an appropriate power of t multiplied to Z(t), we have: vol(U) → lim tn/2Z(t) = , (1.0.6) t 0+ (4π)n/2

which is equivalent to Weyl’s→ law (see [85, Thm. 8.16, p. 115]). When S. Minakshisundaram and Å. Pleijel proved Weyl’s law for closed Riemannian manifolds, they showed that the heat trace has an asymptotic2 expansion 1 Z(t) ∼ (a + a t + a t2 + ··· ) (1.0.7) (4πt)n/2 0 1 2 for t 0+, and that a0 = vol(M), where→M is the underlying manifold. Later, H. P. McKean Jr. and I. M. Singer [74] calculated (in addition to a0) a1 and a2; according to their result, 1 a = S, (1.0.8) 1 6 ZM where S is the scalar curvature of M; the coefficient a2 is not as simple as the earlier ones, but it is the integral of a polynomial of degree 2 in the components of the Riemann curvature tensor with respect to an orthonormal framing (the polynomial is invariant under the choice of the framing). The higher order coefficients are, in general, extremely hard to calculate. What is interesting is that, if M is 2-dimensional, then, applying the Gauss-Bonnet3 theorem [15, 45] to the result 1.0.8 gives us π a = χ(M), 1 3 where χ(M) is the Euler characteristic of M. Thus, even though the asymptotic expansion 1.0.7 for Z(t) is, a priori, determined by the analytical properties of the Laplacian, the coefficient a1 is completely determined by the topology of M and is independent of the metric.

LOCAL INDEX THEOREM. The story becomes even more interesting as we consider the Dirac operator, which is the “square root” of the Laplacian. As P. A. M. Dirac [29] found out, the natural vector bundle for the Dirac operator is the spinor bundle (whose ﬁbers are irre-

2 The asymptotic expansion 1.0.7 means that, for each nonnegative integer N, we n/2 N k N have (4πt) Z(t) − k=0 akt = o(t ) for t 0+. 3 O. Bonnet [15] writes that the theorem had also been proved by J. Binet. P →

3 ducible modules over the Clifford algebra generated by the tangent space). Suppose D/ is a Dirac operator on the sections of the spinor bundle S over a compact even-dimensional Riemannian manifold M. By the representation theory of Clifford algebras, the spinor bundle, in this case, is naturally bi-graded and so is its space of sections: Γ(S) = Γ(S)+ ⊕ Γ(S)−. In the most interesting cases, the Dirac operator is an odd operator: 0 D/ D/ = − . D/ + 0 Moreover, its spectrum is an unbounded discrete subset of iR, each eigenvalue occurring with ﬁnite multiplicity. The space Γ 2(S) of square-integrable sections of S admits a Hilbert space direct sum decomposition into the eigenspaces of D/ . (We shall review all these 2 in detail in Chapter 8.) Now consider the operator etD/ , t ∈ ]0, [, and its super-trace, that is, the ordinary trace over the even domain minus the trace over the odd domain: ∞ / 2 / / / / Str(etD ) = tr(etD−D+ ) − tr(etD+D− ). It turns out that there is a “super-symmetry” among the nonzero 2 eigenvalues of D/ in that the multiplicity in the even domain is equal 2 to that in the odd domain; as a result, the super-trace of etD/ is equal to the difference between the dimension of the kernel of D/ in the even domain and that in the odd domain; this quantity is what is called the graded index of D/ :

Inds D/ = dim ker(D/ +) − dim ker(D/ −). In short, tD/ 2 Inds D/ = Str(e ). This is called the McKean-Singer formula, for such a phenomenon was ﬁrst observed by H. P. McKean Jr. and I. M. Singer in their above mentioned work regarding the asymptotic expansion of the heat 2 trace. The operator etD/ admits an integral kernel, that is, a function (t, x, y) 7 kt(x, y) ∈ End(Sy,Sx) for t ∈ ]0, [ and (x, y) ∈ M × M such that → tD/ 2 ∞ (e σ)(x) = kt(x, y)σ(y) voly ZM for σ in Γ(S). Here voly is the Riemannian volume form of M at y. In terms of the integral kernel,

tD/ 2 Inds D/ = Str(e ) = Str(kt(x, x)) volx. ZM

4 Because of this, Str(kt(x, x)) volx is called the index density of D/ . R. T. Seeley’s work [87] shows that kt(x, x) admits an asymptotic expansion of the form 1 k (x, x) ∼ (a (x) + a (x)t + a (x)t2 + ··· ) t (4πt)n/2 0 1 2 for t 0+, where n = dim(M)/2. Then, 1 Str(k (x, x)) ∼ (Str(a (x))+Str(a (x))t+Str(a (x))t2 +··· ). →t (4πt)n/2 0 1 2 M. Atiyah, R. Bott, and V. K. Patodi [8] demonstrated that the leading nonzero term comes from an/2 and that (in our simple setting)

Str(an/2) vol = Aˆ where Aˆ is the Hirzebruch Aˆ -class on M. Thus, the index density satisﬁes the asymptotic equality

Str(kt(x, x)) vol = Aˆ + O(t), (1.0.9) and thus,

Inds D/ = Aˆ + O(t). ZM Since the left-hand side is independent of t, we have

Inds D/ = A,ˆ (1.0.10) ZM which is the celebrated Atiyah-Singer index theorem [10] in its sim- plest setting. Once again, we have an equation whose one side is analytical in nature while the other side is topological. The Atiyah- Singer index theorem is a far-reaching statement that generalizes the Gauss-Bonnet-Chern theorem [23], the Riemann-Roch-Hirzebruch theorem [57, 82, 84], the Hirzebruch signature theorem [58], and the Lefschetz ﬁxed point theorem [70]; see [7; 12, § 4.1; 85, Ch. 10]. (M. Atiyah’s account on the history leading to the index theorem can be found in [5].) What we have described here is the so-called heat kernel proof of the index theorem. This is not the original approach took by M. Atiyah and I. M. Singer. But the heat kernel proof is stronger in that we have a local index theorem, namely, Equation 1.0.9, which says that the leading nonzero term in the asymptotic expansion of the index density is completely determined by the topology. Because of its local nature, the local index theorem is primed for generaliza- tions to noncompact manifolds.

THE THESIS. Soon after M. Atiyah and I. M. Singer proved the index theorem, R. Bott [16] examined the special case of homogeneous spaces G/K where G is a compact connected Lie group and K is a closed connected subgroup. Using H. Weyl’s theory (integral formula, character formula, and so on) R. Bott was able to verify the

5 index theorem by-passing much of the analytic or topological arguments. Expectation of such simplification for the local index theorem is the motivation behind this dissertation. It seems, however, that the known proofs for the local index theorem — such as [8, 13, 47] — in themselves do not simplify even if we restrict the manifolds under consideration to compact homogeneous spaces. What is common in the aforementioned proofs is that they use the Riemannian Dirac operator (see Section 6.2.6 for definition). The main thesis of this dissertation is then that the use of B. Kostant’s cubic Dirac operator [65] in place of the Riemannian Dirac operator leads to substantial simplifications; A. Alekseev and E. Meinrenken’s quantum Weil algebra [2,3] also plays an important role. To give a quick sketch, take a bi-invariant metric on G; let g, k be the Lie algebras of G and K, respectively, and let p be the orthogonal complement of k in g. A. Alekseev and E. Meinrenken introduced [2] the quantization map Q :(S(g) ⊗ ∧(p))K (U(g) ⊗ Cl(p))K. (1.0.11) Here S(g) and U(g), respectively, are the symmetric algebra and the universal enveloping algebra generated→ by g; ∧(p) and Cl(p), respectively, are the exterior algebra and the Clifford algebra generated by p; the decoration ( · )K denotes the subspace of K-invariants in the respective algebra. The quantization map Q is a vector space isomorphism given by the graded symmetrization of certain generators. Its image (U(g) ⊗ Cl(p))K is called the relative Weil algebra for the pair (g, k). The algebraic structure of the relative Weil algebra sin- gles out an element D/ (g, k), which happens to be B. Kostant’s cubic Dirac operator. (The details will be reviewed in Chapter 5.) What this dissertation demonstrates is then the following. Let E be a finite-dimensional Cl(p)-module, and let Γ(G × E)K be the space of K-equivariant sections of the trivial bundle G × E G. The elements of the relative Weil algebra can be naturally identified K as G-equivariant differential operators on Γ(G × E) . Now→ there is a vector space isomorphism between Γ(G × E)K and the space Γ(E(G)) of sections of a certain4 K-equivariant bundle E(G) G/K; and there is a differential operator D/ g/k on Γ(E(G)) such that the following diagram commutes: → D/ (g,k) Γ(G × E)K / Γ(G × E)K ∼ ∼

Γ(E(G)) / Γ(E(G)) D/ g/k

4 Because the inner product on g is bi-invariant, the adjoint representation of k on p is antisymmetric. Hence, we have a Lie algebra homomorphism k so(p) ' spin(p) ⊆ Cl(p). This induces a K-action on E. The bundle E(G) is then the K-orbit −1 space of G × E under the right K-action (g, v) · k = (gk, k · v). Details→ of this construction will be reviewed in Sections 5.1 and 6.1.

6 This operator D/ g/k is a Dirac operator on G/K, but it is not the Rie- mannian Dirac operator which was used, for instance, in E. Getzler’s proof [47] of the local index theorem. Our strategy is to deduce 2 2 the heat kernel of D/ g/k via the heat kernel of D/ (g, k) . To that end, we consider the preimage L of D/ (g, k)2 under the quantization map 1.0.11. This preimage has, as an element of (S(g) ⊗ ∧(p))K, a natural identiﬁcation as a differential operator on Γ(g×E)K. We shall show that the algebraic relation Q(L) = D/ (g, k)2 brings about a geometric equation involving the Laplacian ∆g on the Euclidean space g and the exponential chart near the identity of G (Proposition 7.2.11); the equation is so simple that the asymptotic expansion of the heat kernel of D/ (g, k)2 follows immediately. What 2 remains is to deduce from this the heat kernel of D/ g/k. We follow, in outline, the approach of N. Berline and M. Vergne [13]; what differs (aside from the already mentioned use of the cubic Dirac operator) is that the natural principal K-bundle G G/K is used instead of a principal Spin(p)-bundle over G/K; the beneﬁt is that, owing to the homogeneity of G, almost all calculations→ are brought down to the level of Lie algebras, and this further eases the proof.

OUTLINE. We begin by reviewing the theory of compact Lie groups, focusing on the those parts that are relevant to the Laplacian associated to a bi-invariant metric. This is done in Chapter 2. Chapter 3 is meant to be a warm-up with the heat trace of the Laplacian. We verify the volume formula of Harish-Chandra using Weyl’s law and the Euler-Maclaurin formula. We then tackle, in Chapter 4, the asymptotic expansion of the heat kernel of the Laplacian using Lie algebra methods; more precisely, we use the Duflo isomorphism. The efficiency of the Lie- algebraic approach becomes manifest. Chapter 5 gives an extra supply of algebra, namely, the theory of quantum Weil algebra, which is necessary for our Lie-algebraic proof of the local index theorem Chapter 6 shows how the elements of the quantum Weil algebra are naturally identified as equivariant differential operators on Clif- ford module bundles. Having made the necessary preparations, we derive, in Chapter 7, the asymptotic expansion of the heat kernel of the square of the Kostant-Dirac operator using the quantization map between the classical and the quantum Weil algebra. The calculation is a modest update of what is done in Chapter 4. Chapter 8 is the high point of this dissertation where we present a Lie-algebraic proof of the local index theorem for compact homogeneous spaces. In the final chapter, we take a slightly different view point on

7 the index of the Dirac operator, namely, as a distribution associated to a transversally elliptic operator. We prove a pleasant theorem (Theorem 9.2.17) that involves the Duﬂo isomorphism and Chern- Weil theory.

OPEN QUESTIONS. Below are some questions that arise from this work, beginning with the immediate ones:

(1) Can the Lie algebra method be generalized to symmetric spaces of noncompact type? Suppose that G/K is a symmetric space of noncompact type (for instance, a hyperbolic space). The index theory for the Dirac operator on G/K has featured in representation theory since the pioneering work of R. Parthasarathy [77]. M. Atiyah and W. Schmid [9] used Atiyah’s L2-form of the index theorem to give a geometric account of the construction of discrete series representations. In more recent work, V. Lafforgue [67] used the L2-index theorem to show that the Baum-Connes conjecture for G also leads to a geometric description of the discrete series using Dirac operators and harmonic spinors. In both cases, general techniques of analysis and Riemannian geometry are used to establish the index theorem; but it is natural to guess that the Lie-algebraic approach demonstrated in this dissertation should extend to symmetric spaces. A simpliﬁed proof like the one sketched above would considerably simplify the connection between K-theory, Dirac operators, and the discrete series. Some difﬁculties are immediately apparent, such as the fact that the natural Laplacian on G (the Casimir operator) is not elliptic, but there are good indications that they can be overcome.

(2) Does Theorem 9.2.17 hold for general principal bundles? Let K be a compact Lie group, and let P be a principal K-bundle over a compact manifold M. A Dirac operator on M can be lifted — using a connection on P — to a transversally elliptic operator D/ . The deﬁnition of the distributional index of D/ applies to this case. An obvious generalization of Theorem 9.2.17 is then to replace G/K with M in Equation 9.2.18.

(3) Can the Kostant-Dirac operator be used in the proof of the local index theorem in its full generality? Let M be a compact spin manifold of dimension n. Let FrSO(M) be the principal SO(n)-bundle of oriented orthonormal frames for the tangent bundle of M. Let PSpin(M) FrSO(M) be a spin structure for M. (Hence, PSpin(M) is a principal Spin(n)-bundle that is a ﬁber-wise double covering of→ FrSO(M).) The principal bundle PSpin(M) plays a major role in N. Berline and M. Vergne’s proof of the local index theorem. Is there a room for the Kostant-Dirac operator D/ K associated to the compact Lie group

8 K = SO(n)? (For the deﬁnition of D/ K, see Section 5.2.1.) For instance, suppose D/ M is a Dirac operator on M. This can be lifted to a transversally elliptic operator on PSpin(M). A suggestion by N. Higson is to augment D/ M with D/ K so that we would have an elliptic operator; would this streamline N. Berline and M. Vergne’s proof?

(4) What is the relationship between the Getzler symbol map and the quantization map Q? Let M be a compact spin manifold, and let D/ be the Riemannian Dirac operator on some Clifford module bundle S M. E. Getzler, in his proof [47] of the local index theorem, constructed a symbol map from the ﬁl- tered algebra Ψ of→ pseudo-differential operators on Γ(S) to the associated graded algebra of Ψ. When M is a compact homogeneous space G/K, the Getzler symbol map gives a vector space isomorphism σ :(U(g) ⊗ Cl(p))K (S(g) ⊗ Cl(p))K. This map is not the inverse of the quantization map 1.0.11. But is there a simple relation between→ the two maps? If there is, it could lead to a generalization of Q for differential operators on Clifford module bundles over generic compact spin manifolds, with which one can hope to simplify the proof of the local index theorem in its full generality.

(5) Does the index theorem imply the Duflo isomorphism? The restriction of the quantization map 1.0.11 yields the Duflo isomorphism (Sg)g ' Z(g). (This was shown by A. Alekseev and E. Meinrenken [3].) This underlies in our proof of the local index theorem on compact homogeneous spaces. One could ask whether the converse would hold; can we deduce the Duflo isomorphism from the index theorem?

(6) Can the Euler-Maclaurin formula explain the appearance of the Aˆ -class in the index formula? Both the Euler-Maclaurin formula 3.1.1 and the Hirzebruch Aˆ -class involves the power series of x . 1 − e−x Is this a coincidence, or is there a deeper connection between them?

BACKGROUND MATERIALS. The mathematics we take for granted is roughly the following (references indicate the level): algebra, D. S. Dummit and R. M. Foote [31]; functional analysis, W. Rudin [86]; theory of differentiable manifolds, L. W. Tu [91]; differential geometry, S. Kobayashi and K. Nomizu [63]; Lie theory, F. W. Warner [92].

9 REVIEWOFTHETHEORYOF2 COMPACTLIEGROUPS

E review some basic notions surrounding the Laplacian on a W compact Lie group that is endowed with a left-invariant metric. This will also serve as an opportunity to introduce the notation we use throughout this dissertation. For basic Lie theory, we refer to[92, Ch. 3]. For background in differential geometry, see [22, Ch. 1]; the same reference contains (Ch. 3) an excellent account on the differential geometry of homogeneous spaces. For general reference on the representation theory of (locally) compact groups, we refer to [83].

2.1 ANALYTICALASPECTS

2.1.1 Throughout this dissertation, G denotes a compact Lie group unless mentioned otherwise. We shall always denote the identity element of G by e. We denote by `g and rg, respectively, the left and right translations on G by g ∈ G, that is,

`g : G G, x 7 gx, rg : G → G, x →7 xg. → A vector field Xe on G is left-invariant if → `g∗Xe = Xe for all g in G. Here `g∗ denotes the differential of `g. A left-invariant differential form is defined similarly using the pullback homomor- ∗ ∗ phism `g on the space Ω (G) of differential forms on G.. If Xe is a left-invariant vector field, then it is completely deter-

10 mined by its value at the identity, since

Xeg = `g∗Xee. Conversely, a tangent vector X at the identity induces a left-invariant vector ﬁeld Xe on G by deﬁning the value of Xe at g by

Xeg = `g∗X. Therefore, there is a one-to-one correspondence between the space G X(G) of left-invariant vector ﬁelds and the tangent space TeG of G at the identity. The space X(G)G is closed under the commutator [92, Prop. 3.7(c), p. 84]. Deﬁning the bracket operation on TeG as

[X, Y] := [X,e Ye]e makes TeG a Lie algebra. This is known as the Lie algebra of G, and it is customary to denote it by the lowercase black-letter g. For each g in G, let c(g) be the conjugation map G G, h 7 ghg−1. Mapping g to the differential of c(g) at the identity gives the adjoint representation of G: → → Ad : G Aut(g), (2.1.2) g 7 c(g)∗,e. → We shall denote the image of g ∈ G under Ad by Adg. The differential of Ad at the identity, in turn,→ gives the adjoint representation of g: ad : g End(g), X 7 Ad∗,e(X). → We shall denote the image of X ∈ g under ad by adX. It is well-known that → adX(Y) = [X, Y] for X and Y in g. (See, for instance, [92, Prop. 3.47].)

2.1.3 BI-INVARIANT METRICS. Suppose h , i is an inner product on g. Extend this inner product to the whole tangent bundle TG by left translations; that is, for X and Y in TgG, set −1 −1 hX, Yi = h`g∗ X, `g∗ Yi. This deﬁnes a left-invariant metric on G in the sense that

hX, Yi = h`x∗X, `x∗Yi for any x in G. This metric would also be right-invariant if and only if the inner product on g is Ad(G)-invariant; this owes to the following equalities: For X and Y in g,

hrg∗X, rg∗Yi = h`g−1∗ ◦ rg∗X, `g−1∗ ◦ rg∗Yi = hAdg−1 X, Adg−1 Yi. So we see that there is a one-to-one correspondence between the Ad(G)-invariant inner products on g and the bi-invariant (that is, left- and right-invariant) metrics on G. Henceforth, we shall say

11 that an inner product h , i on g is invariant if the Ad(G)-action is symmetric or the ad(g)-action is antisymmetric with respect to h , i. For compact Lie groups, we can always make an inner product on g invariant by replacing (if necessary) the original inner product with its average over G:

hhX, Yii := hAdg X, Adg Yi dg. ZG Here dg is a Haar measure of our choice on G. From now on we shall always assume that our compact Lie group G is endowed with a bi-invariant metric h , i.

2.1.4 THE KILLING FORM. There is a natural bilinear form on the Lie algebra g, namely, the Killing form, κ : g × g C, (X, Y) 7 tr(adX ◦ adY). → Owing to the fact that adAdg(X) = Adg ◦ adX ◦ Adg−1 , we have the symmetry → κ(Adg X, Y) = κ(X, Adg−1 Y). Differentiating both sides of this equation with respect to g gives

κ(adZ X, Y) = −κ(X, adZ Y). Lie groups (not necessarily compact) whose Killing form is non- degenerate are precisely the semisimple ones. If G is compact and semisimple, then the Killing form is negative deﬁnite (for a proof, see [54, Prop. 6.6]). So, for compact semisimple Lie groups, we may use −κ( , ) = h , iκ as the inner product for g. This is an invariant inner product, hence, yielding a bi-invariant metric on G.

2.1.5 RIEMANNIAN VOLUME FORM. Let X1,...,Xn be a orthonormal basis for g. Let θ1, . . . , θn be the dual basis for g∗. Extending the form θ1 ∧ ··· ∧ θn ∈ ∧ng∗ to all of G by left translations, we obtain a bi-invariant volume form on G. This is identical to the Riemannian1 volume form of G. Every left-invariant volume form on G is completely determined by its value at the identity; so every left-invariant volume form on G is a scalar multiple of the Riemannian volume form, and hence, they are all bi-invariant. Henceforth, we shall speak of an invariant volume form without reference to left or right invariance.

1 Suppose M is a Riemannian manifold with metric h , i. For a local coordinate system (x1, . . . , xn), let η denote the matrix whose (i, j)-entry is ηij = h∂i, ∂ji where ij −1 1/2 1 ∂i = ∂/∂xi. Let η denote the (i, j)-entry of η . Let vol := | det(η)| dx ∧ ··· ∧ dxn, where {dx1, . . . , dxn} is the dual basis for the cotangent space with respect to ∂1, . . . , ∂n. This local expression for vol deﬁnes a global n-form called the Riemannian volume form of M.

12 A similar statement can be made for invariant measures. Let C(G) be the space of continuous functions on G.A left Haar measure on G is a positive linear functional Υ : C(G) R that is left ∗ ∗ invariant in the sense that Υ(f) = Υ(`gf) for all g in G, where `gf is the pullback of f along the left translation `g.A right→ Haar measure is deﬁned similarly using the right translations. An invariant volume deﬁnes a left Haar measure that is also a right Haar measure. Be- cause a left Haar measure on G is unique up to a constant factor (see [61, Thm. 6.8]), there is a one-to-one correspondence between invariant volume forms and left Haar measures on G. Thus, every left Haar measure on G must be a right Haar measure, which is to say that G is unimodular.

2.1.6 THE LAPLACIAN. The Laplacian is defined as follows. (The definition is appropriate for any Riemannian manifold.) Let C (G) be the space of smooth functions on G, and let X(G) denote∞ the space of smooth vector fields on G. We define the gradient operator grad : C (G) X(G) and the divergence operator div : X(G)

C (G) by∞ ∞ → hgrad f, Xi = Xf, →

(div X) vol = LX vol, where vol is the Riemannian volume form on G and LX is the Lie derivative with respect to X. The Laplacian (or the Laplace-Beltrami operator) ∆G : C (G) C (G) is deﬁned by

∞ ∆G∞f := div(grad f). → The deﬁnition of ∆G depends only on the metric. Because our metric h , i is bi-invariant, so is the Laplacian in the sense that ∆G commutes ∗ ∗ with `g and rg for all g in G. An expression for the Laplacian in local coordinates (x1, . . . , xn) can be given as follows. (See [55, Ch. II, § 2] for details.) Let η be ij the matrix deﬁned by ηij = h∂i, ∂ji where ∂i = ∂/∂xi. Let η denote the (i, j)th entry of η−1. Then,

1 1/2 ij ∆Gf = ∂i(| det(η)| η ∂jf). (2.1.7) | det(η)|1/2 i,j X

2.1.8 THE SPECTRUM OF THE LAPLACIAN. The Laplacian ∆G is de- ﬁned over C (G), which is a dense subspace of L2(G). The domain 2 can be extended∞ to the Sobolev space H (G), that is, the space of L2-functions u on G such that all distributional 1st and 2nd deriva- 2 2 tives lie in L (G). This extension, which we denote by ∆G : H (G) 2 2 L (G), is the unique self-adjoint extension of ∆G : C (G) L (G).

Thus, in the language of unbounded operator theory,∞ the Laplacian→ is essentially self-adjoint on C (G). For details on these matters,→ see [85, Ch. 5]. ∞

13 Let 1 denote the inclusion map H2(G) , L2(G). The operator 1−∆G admits an inverse that is compact (see [89, § 5.1]). So, by the −1 Spectral theorem, the eigenfunctions { uk }k=→1 of (1 − ∆G) form an 2 orthonormal basis for L (G). They are also∞ eigenfunctions of −∆G. Owing to the regularity of elliptic differential operators, the eigenfunctions are of class C . We can also conclude from the Spectral theorem that the eigenfunctions∞ uk can be ordered in such a way that the corresponding eigenvalues −λk of ∆G form a nonincreasing unbounded sequence of negative real numbers,

0 > −λ1 > −λ2 > −λ3 > ··· .

2.1.9 THE HEAT DIFFUSION OPERATOR. The heat diffusion operator of ∆G is deﬁned as

 e−tλ1  e−tλ2 t∆G e :=  e−tλ3  . .. for t > 0, where the matrix on the right-hand side is with respect to the basis consisting of the eigenfunctions of the Laplacian. The heat diffusion operator is an integral operator with a C - kernel. (This owes to the Schwartz kernel theorem; see [89, Ch.∞ 4, § 6, p. 345] or [85, Prop. 5.31, p. 83].) This means that there is some C -function (t, x, y) 7 Kt(x, y) on ]0, [ × G × G such that

∞ t∆G (e f)(x→) = Kt(x, y)f(y∞) voly (2.1.10) ZG 2 for any f in L (G); here voly is the Riemannian volume form at y. The kernel Kt is called the heat kernel of ∆G. It has the following fundamental properties: (i) It satisﬁes the heat equation

(∂t − ∆G)Kt(x, y) = 0, where the Laplacian applies to the ﬁrst variable x. (ii) As t 0+, it converges to the Dirac delta distribution δ(x − y) in the sense that, for any smooth function f on G, → Kt(x, y)f(y) voly f(x) ZG uniformly. →

The smooth kernel Kt is uniquely determined by the above two properties; see [85, Prop. 7.5, p. 96]. Recall that ∆G is bi-invariant; in particular, we have ∗ ∗ ∆G = `g ◦ ∆G ◦ `g−1 for any g in G, and thus,

t∆G ∗ t∆G ∗ (e f) = `g ◦ e ◦ `g−1 .

14 This means, in terms of the heat kernel,

−1 Kt(x, y)f(y) voly = Kt(gx, y)f(g y) voly ZG ZG for any f in L2(G). Substituting gy for y in the right-hand side and using the invariance of the volume form, this equation can be restated as

Kt(x, y)f(y) voly = Kt(gx, gy)f(y) voly. ZG ZG This implies Kt(x, y) = Kt(gx, gy). Substituting g with x−1 gives us −1 Kt(x, y) = Kt(e, x y). Hence, the heat kernel is completely determined by the function

kt : y 7 Kt(e, y). (2.1.11)

We shall call this the heat convolution kernel of ∆G. With it, Equa- tion 2.1.10 can be rephrased as →

t∆G −1 (e f)(x) = kt(x y)f(y) voly. (2.1.12) ZG Substituting xy for y and using the invariance of the volume form, we get

t∆G (e f)(x) = kt(y)f(xy) voly. (2.1.13) ZG A consequence of et∆G being an integral operator with a C - kernel is that it has a ﬁnite trace. The trace of the heat diffusion∞ operator, Z(t) = tr(et∆G ), is called the partition function or the heat- trace of ∆G. It can be calculated in terms of the heat kernel as follows (see [85, Thm. 8.10, p. 113]):

Z(t) = Kt(x, x) volx = kt(e) vol(G). (2.1.14) ZG Since a Riemannian manifold is locally “ﬂat”, it is reasonable to expect that the heat convolution kernel, which governs the heat diffusion, looks almost Gaussian for small time t near e. We seek a formal solution of the form 2 ket(x) := ht(x) a0(x) + a1(x)t + a2(x)t + ··· , (2.1.15) where ht is the Gaussian kernel: e−d(x)2/4t h (x) = . (2.1.16) t (4πt)dim M/2

Here d(x) denotes the distance from e to x. Write ket = htst where : i st = i=0 ait . Then, on a neighborhood U of e where log : U g, P∞ → 15 −1 x 7 exp (x), is well-defined, the heat equation (∂t + ∆G)ket = 0 can be rewritten as (see [85, Eq. 7.16, p. 102] or [12, Prop. 2.24, p.→ 82]) 1 1 ht ∂t + Re + ˜ ◦ ∆G ◦ st = 0, (2.1.17) t ˜ where Re is the radial vector field in normal coordinates (this means that the pushforward of Rex along log is equal to X := log(x) as we identify the tangent space TXg with g) and ˜ = j ◦ log with sinh ad /2 j(X) = det1/2 X . adX /2 Setting each coefficients of powers of t in Equation 2.1.17 as 0, we obtain a family of differential equations:

Rae 0 = 0, ai−1 (Re + n)ai = −˜∆G , i 1. ˜ >

This can be solved inductively under the condition a0(e) = 1. The formal power series 2.1.15 is called the asymptotic heat kernel or the asymptotic expansion for the true heat kernel kt because of the following property: Let r be any nonnegative integer, and let k · kCr denote the usual norm2 on the space Cr(G) of Cr-functions on G; then, for each nonnegative integer n there is a positive integer N such that, for each m > N, m i n kt − ht ait 6 C|t| Cr i=0 X for sufﬁciently small t; here C is a constant that depends on n, m, and r. For details, see [85, Thm. 7.15, p. 101] or [12, Thm. 2.26, p. 83, Thm. 2.29, p. 85].

2.1.18 In general, a function f :]0, [ E, where E is a Banach space, is said to be asymptotically equal to the formal sum i=0 ai of functions ai :]0, [ E if, for each∞ positive→ integer n there∞ is a positive integer N such that P ∞ → m n f(t) − ai(t) = O(t ) i=0 X holds for all m > N. (The above equation means that there is some m n constant C such that kf(t)− i=0 ai(t)k 6 C|t | for sufﬁciently small

2 d Let d = dim M and let α = (α1,P . . . , αd) denote a multi-index in N . Then the r r α d standard C -norm of f in C (M) is kfkCr = sup k∂ fksup where |α| := αi, |α|6r i=1 ∂α1 ∂αd α : : ∂ = α1 ··· αd , and kfksup = supx∈M |f(x)|. P ∂x1 ∂xd

16 t.) If this is the case, we write

f ∼ ai, ∞ i=0 X and call i=0 ai an asymptotic expansion for f. In the special case n where f ∼ ∞0, that is, f(t) = O(t ) for all positive integer n, we shall sayP that f is of order t or write f = O(t ). We say that g :]0, [ E is asymptotically∞ equal to f and write∞ f ∼ g, ∞ → if f − g is asymptotically equal to 0.

2.2 ALGEBRAICASPECTS

2.2.1 We pointed out earlier that the Laplacian is bi-invariant in that ∗ ∗ it commutes with `g and rg for all g ∈ G. In general, a differential operator D on C (G) is said to be (left) invariant if

∞ ∗ ∗ D = `g ◦ D ◦ `g−1 for all g in G. It is said to be bi-invariant if it furthermore satisﬁes ∗ ∗ D = rg ◦ D ◦ rg−1 . We shall denote by D(G) the algebra of invariant differential operators on C (G). Assuming G is connected, the subalgebra of bi- invariant operators∞ is precisely the center of D(G). But before we review why this is so, let us brieﬂy go over the terminologies and notations of the representation theory that we shall use.

2.2.2 A representation of a compact Lie group G is a continuous group homomorphism u : G Aut(V) where V is a topological vector space. (If V is ﬁnite-dimensional, then a continuous representation is automatically of class →C because every continuous group homomorphism of Lie groups are∞ of class C ; see [92, Thm. 3.39, p. 109].) Most of the time, we shall be interested∞ in the unitary representations of G; that means the vector space V, called the representation space of u, is a Hilbert space and that the representation u maps G into the space of unitary operators on V. The unitarity of the representation is equivalent to the invariance of the inner product under the action of G; this owes to the following equality: hu(g)(v)|u(g)(w)i = hu(g)∗u(g)(v)|wi, for g ∈ G and (v, w) ∈ V × V. It turns out that any representation of a compact Lie group on a Hilbert space is equivalent to a unitary representation; indeed, for any inner product h , i on V, the : averaged inner product hhv, wii = Ghu(g)v, u(g)wi dg is an invariant inner product on the same Hilbert space, and the norm associ- R ated to the new inner product is equivalent to the original one (see

17 [83, Prop. 2.2, p. 14]). If we have a representation u : G Aut(V), then we say that V is a G-vector space. The dimension of V is referred to as the dimension of the representation. The representation→ is said to be real or complex according to the ﬁeld of scalars of V. For our purposes, we shall mainly consider the complex representations.

2.2.3 NOTATION. Let u : G Aut(V) be a representation. For g ∈ G and v ∈ V, we shall often write u(g)(v) as ug(v), or even as g · v if there is no risk of confusion.→

2.2.4 A linear map T : V W between two G-vector spaces is a G-map (or an intertwiner) if it is G-equivariant, that is, g · T(v) = T(g · v) holds for all g ∈ G→and all v ∈ V. The space of G-maps V W is denoted by HomG(V, W). If the G-map T : V W is homeomorphic, then we say that V and W are G-isomorphic and that→ the representations on V and W are equivalent. →

2.2.5 Let u : G Aut(V) be a ﬁnite-dimensional representation. Its character is the function

→ χu : G k, g 7 tr(ug), where k is the ﬁeld of scalars of V→. → The induced Lie algebra representation u∗ : g End(V) associated to u is deﬁned as the differential of u at the identity. This means that, for X in g, →

d u∗(X)(v) = u(exp tX)(v). dt 0 Put in another way, eu∗(X) = u(exp(X)); (2.2.6) see [92, Thm. 3.32, p. 104]. The dual (or contragredient) representation induced by u is the representationu ˇ : G Aut(V∗) defined by −1 (uˇgφ)(v) = φ(ug (v)) (2.2.7) → where φ ∈ V∗ and v ∈ V. If u0 : G Aut(V 0) is another finite-dimensional representation, then the direct sum and the tensor product representation u ⊕ u0 : 0 0 0 G Aut(V →⊕ V ) and u ⊗ u : G Aut(V ⊗ V ), respectively, are defined by → 0 0 → 0 0 (u ⊕ u )g(v ⊕ v ) = ugv ⊕ ugv , 0 0 0 0 (u ⊗ u )g(v ⊗ v ) = ugv ⊗ ugv .

2.2.8 SCHUR’S LEMMA. A simple yet very useful theorem in representation theory is Schur’s lemma. There are several versions of

18 Schur’s lemma, one of which is as follows: Let V and W be ﬁnite- dimensional vector spaces over C (or any algebraically closed ﬁeld). Suppose there is a family of pairs of linear maps,

{ (φa, ψa) ∈ End(V) × End(W) }a∈A, with respect to which the spaces V and W are irreducible, that is, they admit no proper nonzero subspaces that are invariant under { φa }a∈A and { ψa }a∈A, respectively. Then the space of intertwiners,

{ T : V W | T ◦ φa = ψa ◦ T, ∀a ∈ A }, is 1-dimensional or 0-dimensional according to whether dim V is equal to dim W. For a→ proof, see [99, Lem. 3.1.C, p. 83].

2.2.9 IRREDUCIBILITY. Let u : G Aut(V) be a representation. A subspace W of V is said to be invariant if ug(W) ⊆ W for all g in G. If V admits a closed invariant proper→ subspace other than {0}, then we say that the representation is reducible; otherwise, the representation is said to be irreducible. A reducible representation is said to be completely reducible if the representation space is isomorphic to a direct sum of irreducible subspaces (that is, a Hilbert space direct sum, if the representation is on a Hilbert space). Suppose the representation u is a unitary representation on a Hilbert space. If W is an invariant subspace, then its orthogonal complement W⊥ is also invariant, owing to the invariance of the inner product. Thus, any representation of a compact Lie group on a finite-dimensional space or an infinite-dimensional Hilbert space, is reducible if and only if it is completely reducible (by Zorn’s lemma). It turns out that an irreducible representation of a compact Lie group on a complex Hilbert space must be finite-dimensional (see [18, Thm. 4.3, p. 25]). Hence, any complex representation of a compact Lie group on a Hilbert space is a direct sum of finite-dimensional irreducible subspaces. The complete set of distinct (that is, non-equivalent) irreducible unitary representations of G is called the unitary dual of G; we shall denote it by Gb. For each u in Gb, the u-isotypic component of a G-vector space V is defined as the maximal subspace W of V on which the G-action is isomorphic to a direct sum of copies of u; owing to Schur’s lemma, we have W ' HomG(U, V) ⊗ U, where U is the representation space of u. The dimension of the space HomG(U, V) is called the multiplicity of u in V.

2.2.10 THE UNIVERSAL ENVELOPING ALGEBRA. The left translations on G induces a natural G-action on C (G); the left-regular action of

∞

19 g ∈ G on f ∈ C (G) is deﬁned by ∞ (L(g)f)(x) = f(g−1x). There is also the right-regular action induced by the right translations: (R(g)f)(x) = f(xg). The induced Lie algebra representation of X in g associated to the right-regular representation is given by

d (R∗(X)f)(x) = f(x exp(tX)). dt 0 Note that this action is exactly that of the left-invariant vector ﬁeld Xe on G generated by X. Thus,

(R∗(X)f)(x) = (Xfe )(x). Hence, we have a map τ : g D(G), X 7 X.e → Because []X, Y] = [X, Y] (where the bracket on the right is just the e e → commutator in D(G)) the map τ extends to the universal enveloping algebra U(g) of g, which is the algebra constructed by first taking the tensor algebra T(g) of g and then taking the quotient by the ideal J(g) generated by the elements of the form X ⊗ Y − Y ⊗ X − [X, Y]. The extended map τ : U(g) D(G), (2.2.11) X1 ⊗ · · · ⊗ Xk + J(g) 7 Xe1 ◦ · · · ◦ Xek, is an algebra isomorphism (see [54→, Ch. II, Prop. 1.9]). It is custom- → ary to write X1 ⊗ · · · ⊗ Xk + J(g) in U(g) as X1 ··· Xk and Xe1 ◦ · · · ◦ Xek in D(G) as Xe1 ··· Xek. Let us verify the claim we made in Section 2.2.1 that, if G is connected, the center of D(G) is the subalgebra of bi-invariant differential operators. The argument we give here is from [55, p. 283]. We need to show that an invariant differential operator D commutes ∗ ∗ with every operator in D(G) if and only if rg ◦D◦rg−1 = D for all g in G. As we can see from the isomorphism 2.2.11, the elements of the form Xe1 ··· Xek span D(G); hence, we may assume that D = Xe1 ··· Xek. Moreover, the left-invariant vector fields generate D(G); so it is sufficient to check that the commutator [Y,De ] := YDe − DYe is zero for ∗ ∗ any Y in g if and only if rg ◦ D ◦ rg−1 = D for all g in G. For the “only if” assertion, start by noting that

∗ ∗ d −1 d (rg ◦ Xe ◦ rg−1 f)(x) = f(xg exp(tX)g ) = f(x exp(t Adg X)) dt 0 dt 0

20 for a smooth function f on G. Hence,

∗ ∗ rg ◦ Xe ◦ rg−1 = Ad^g(X). (2.2.12) ad If g = exp(Y), then Adg(X) = e Y (X). So

ad Ad^g(X) = e Ye(Xe), G G ad n where ad : X(G) X(G) , X 7 [Y, X], and e Ye = ad /n! Ye e e e n=0 Ye (the convergence is not an issue since X(G)G is ﬁnite-dimensional). P∞ The domain of ad extends, as an inner derivation, to D(G). Then, Ye → → we have ad (D) = [Y,D]. Ye e Note that ad (D) is of order at most k. So the set { adn(D): n ∈ } Ye Ye N is contained in the ﬁnite-dimensional subspace of D(G) consisting of ad n operators of order at most k, and hence, e Ye(D) := ad (D)/n! n=0 Ye is convergent. Now P∞ ∗ ∗ ∗ ∗ ∗ ∗ rg ◦ (Xe1 ··· Xek) ◦ rg−1 = (rg ◦ Xe1 ◦ rg−1 ) ◦ · · · ◦ (rg ◦ Xek ◦ rg−1 )

= Ad^g(X1) ··· Ad^g(Xk), where we have used Equation 2.2.12 for the last equality. So, for g = exp(Y),

∗ ∗ adY adY rg ◦ (Xe1 ··· Xek) ◦ rg−1 = e e(Xe1) ··· e e(Xek)

n1 nk ad (Xe1) ad (Xek) = Ye ··· Ye ∞ n1! ∞ nk! n =0 n =0 X1 Xk n1 nk ad (Xe1) ··· ad (Xek) = Ye Ye . ∞ n1! ··· nk! N=0 n +···+n =N X 1 Xk (2.2.13) Since ad is a derivation, we have. Ye N! N(X ··· X ) = n1 (X ) ··· nk (X ). adY e1 ek ad e1 ad ek e n1! ··· nk! Ye Ye n +···+n =N 1 Xk Thus, Equation 2.2.13 can be rewritten as N ad (Xe1 ··· Xek) ∗ ∗ adY rg ◦ (Xe1 ··· Xek) ◦ rg−1 = = e e(Xe1 ··· Xek). ∞ N! N=0 X ∗ ∗ Therefore, if [Y,De ] = 0 for all Y in g, then rg ◦ D ◦ rg−1 = D for all g in the image of the exponential map. But, since G is connected ∗ ∗ by hypothesis, we may conclude that rg ◦ D ◦ rg−1 = D holds for all g in G; this is because3, for a neighborhood U of e that is in the

3 We could have, at this point, referred to the fact that the exponential map of a compact connected Lie group is surjective (see [18, Thm. 16.3, p. 103]); but the argument presented here works for any connected Lie group.

21 image of the exponential map, any element of G can be expressed as a product of ﬁnite number of elements in U (see [92, Thm. 3.18, p.93]). Conversely, suppose D is bi-invariant; then ∗ ∗ rg ◦ D = D ◦ rg for all g in G. So, for a smooth function f on G, ∗ ∗ (rg(Xe1 ··· Xekf))(x) = (Xe1 ··· Xek(rgf))(x). This means that

d d ··· f(xg exp(t1X1) ··· exp(tkXk)) dt1 0 dtk 0

d d = ··· f(x exp(t1X1) ··· exp(tkXk)g). dt1 0 dtk 0 Substitute exp(sY) for g (where s ∈ R and Y ∈ g), and take the derivative with respect to s at s = 0; we get

d d d ··· f(x exp(sY) exp(t1X1) ··· exp(tkXk)) ds 0 dt1 0 dtk 0

d d d = ··· f(x exp(t1X1) ··· exp(tkXk) exp(sY)). dt1 0 dtk 0 ds 0 This implies YeXe1 ··· Xekf = Xe1 ··· XekYf,e which proves that [Y,De ] = 0 for all Y in g.

2.2.14 THE CASIMIR ELEMENT. What is the element in U(g) that corresponds to the Laplacian under the identiﬁcation of U(G) with D(G)? We claim that it is the (quadratic) Casimir element Ω, which is deﬁned as follows. Let End(g) denote the space of linear maps from g into itself. Let g∗ be the dual space of g. Using the inner product h , i on g, we can construct the isomorphisms End(g) ' g ⊗ g∗ ' g ⊗ g. Since g ⊗ g is a subspace of the tensor algebra T(g), we have a G- equivariant map /J(g) End(g) , T(g) −−− U(g), (2.2.15) where /J(g) denotes the quotient map with respect to the ideal J(g) described in Section 2.2.10.→ The image→ of the identity map on g under the above composition is the Casimir element Ω. Because the Casimir element originated from the identity map, it is in the center

22 Z(g) of the universal enveloping algebra.4 The definition of Ω does not depend on the particular basis for g. But it has a simple expression in terms of an orthonormal basis ∗ X1,...,Xn for g. Let θ1, . . . , θn be the dual basis for g . Then the n ∗ identity map in End(g) corresponds to i=1 θi ⊗ Xi in g ⊗ g. So the Casimir element is n P Ω = XiXi. (2.2.16) i=1 X So why is this the element that corresponds to the Laplacian under the identification of U(g) with D(G)? Recall that the exponential map exp : g G maps a neighborhood U of 0 in g diffeomorphically onto a neighborhood V of e in G. This provides us the exponential chart on V;→ the coordinates (y1, . . . , yn) for g in V are the compo- −1 nents of X = exp (g) with respect to the basis X1,...,Xn; in other n words, g = exp i=1 yiXi . The exponential map we use here is the Lie-theoretic exponential map. There is also the Riemannian exponential mapP Exp : g G coming from differential geometry, which is also a local diffeomorphism near 0 in g. (For more on the Riemannian exponential→ map, see [54, Ch. 1, § 6]). The Rie- mannian exponential map depends on the metric, whereas the Lie- theoretic exponential map has nothing to do with the metric. But the two exponential maps agree if the metric is bi-invariant. This can be seen as follows. Let ∇ be the Riemannian connection so that ∇ hY, Zi = h∇ Y, Zi + hY, ∇ Zi. Using the identity (see [22, p. 2]) Xe e e Xe e e e Xe e 2h∇ Y, Zi = XhY, Zi + YhZ, Xi − ZhX, Yi Xe e e e e e e e e e e e + h[X,e Ye], Zei − h[Y,e Ze], Xei + h[Z,e Xe], Yei and the antisymmetricity of the ad(g)-action, one can check that 1 ∇ Ye = [X,e Ye] (2.2.17) Xe 2 for all X and Y in g. In particular, ∇ X = 0. It follows that the Xe e 0 geodesic γX(t), such that γX(0) = e and γX(0) = X, is a group homomorphism R G (see [54, Ch. 2, Prop. 1.4]). By the uniqueness of one-parameter subgroups, we have γX(t) = exp(tX). This implies that the Riemannian→ exponential map is identical to the Lie- theoretic exponential map. Hence, the matrix [ηij] of the metric under the exponential chart satisfies ηij(e) = δij (Kronecker delta) and ∂kηij(e) = 0. Therefore, the Laplacian at e takes the form

4 The Lie group G acts on g by the adjoint action 2.1.2. This extends to the tensor algebra T(g) by deﬁning g · (Y1 ⊗ · · · ⊗ Yk) = (g · Y1) ⊗ · · · ⊗ (g · Yk) for g in G and vectors Y1,...,Yk in g. This descends to a G-action on U(g). The action of G on A ∈ End(g) is given by (g · A)(X) = g · A(g−1 · X). The composition 2.2.15, under these actions, is G-equivariant. This owes mostly to the fact that the inner product h , i is invariant. The identity map in End(g) is obviously invariant under the G- action. So the Casimir element Ω is a G-invariant element of U(g); this implies that Ω is in the center of U(g).

23 2 ∂i = XiXi. Since the Laplacian is left-invariant, we have

P P ∆G = XeiXei on G. This proves that X τ(Ω) = ∆G (2.2.18) where τ is the algebra isomorphism 2.2.11.

2.2.19 PETER-WEYL THEOREM. Let C(G, C) denote the space of continuous C-valued functions on G. A function f in C(G, C) is said to be G-finite if its orbit under the left-regular action of G spans a finite- dimensional subspace of C(G, C). Denote the subspace of C(G, C) that consists of G-finite elements by C(G, C)fin. The Peter-Weyl theorem [79] says that the space C(G, C)fin of G-finite functions is dense 2 in the space L (G, C) of square-integrable C-valued functions on G. (The theorem is actually for any compact topological group G.) All G-finite functions arise from finite-dimensional unitary representations over C. Let us see why this is so. Suppose u : G Aut(V) is a finite-dimensional unitary representation over C.A representative function of u is a function of the form → G C, −1 g 7 hw, g · viV , → where w and v are vectors in V and h , iV is the inner product on → a complex vector space V. Let Mu be the space of representative functions of u. Then we have a linear isomorphism ∗ Ψu : Mu Vu ⊗ Vu, −1 ∗ (2.2.20) f(g) = hv1|g · v2iV 7 v1 ⊗ v2, ∗ → where v1 is the linear functional defined by w 7 hv1|wi. This is in fact an isomorphism of (G × G)-vector→ spaces, where (g, h) ∈ G × G ∗ ∗ acts on f ∈ Mu by (g, h) · f = LgRhf, and on v1→⊗ v2 ∈ Vu ⊗ Vu by ∗ ∗ (g, h) · (v1 ⊗ v2) = uˇgv1 ⊗ uhv2. This shows that Mu ⊆ C(G, C)fin. But every G-finite function must arise in this way for the following reason. If f is a G-finite function, then f is an element of the finite- dimensional vector space V spanned by the orbit of f under the left- regular action of G. Let φ be the linear functional on V given by evaluating the functions in V at the identity. Then f(g) = φ(g−1 · f). This shows that f is a representative function. Note that the representative functions are smooth. Hence, the continuous functions that are G-finite are of class C . So let us write C(G, C)fin as C (G, C)fin. ∞ To sum up, we have∞ M M ∗ C (G, C)fin = Mu ' Vu ⊗ Vu, (2.2.21) ∞ u∈Gb u∈Gb

24 where Gb is the unitary dual of G. Then the Peter-Weyl theorem can be stated as 2 Mˆ ∗ L (G, C) ' Vu ⊗ Vu, (2.2.22) u∈Gb where Lˆ denotes the Hilbert space direct sum; the orthogonality of this direct sum is known as Schur Orthogonality (see [18, Thm. 2.3, p. 11]). What follows from the isomorphism 2.2.22 is that the subspace 2 2 Lcl(G, C) of L (G, C) that consists of the class functions (that is, functions that are invariant under the conjugation action of G on itself) 2 in L (G, C) is spanned by the irreducible characters of G:

2 Mˆ Lcl(G, C) ' C · χu. (2.2.23) u∈Gb

2.2.24 Assume that the compact Lie group G is connected. Consider the following diagram:

∆G C (G, C)ﬁn / C (G, C)ﬁn

∞ Ω ∞ C (G, C)ﬁn / C (G, C)ﬁn

∼ (2.2.25) (2.2∞.21) ∞ (2.2.21) ∼ L ∗ Ω L ∗ Vu ⊗ Vu / Vu ⊗ Vu u∈Gb u∈Gb The top horizontal map is given by the canonical action of the Lapla- cian on smooth functions. (The representative functions The mid- dle horizontal map is given by the Casimir element via the U(g)- action that is induced (through the algebra isomorphism 2.2.11) by the canonical D(G)-action on C (G, C)fin; as we have seen in Sec- tion 2.2.10, this U(g)-action on ∞C (G, C)fin coincides with that induced by the right-regular action∞ of G on C (G, C)fin. Now, under the vector space isomorphism 2.2.21, the action∞ of G on the compo- ∗ nent Vu ⊗ Vu that is compatible with the right-regular action of G on C (G, C)fin is by 1 ⊗ u, where 1 denotes the trivial representa- ∗ tion of∞ G on Vu. Thus, the bottom horizontal map is given by the Casimir element via the U(g)-action obtained by extending the induced Lie algebra representation u∗ on Vu to U(g). With the above descriptions, the diagram 2.2.25 is commutative. Owing to the connectedness of G, the induced Lie algebra rep- 5 resentation u∗ on Vu is irreducible. Hence, Vu is irreducible under

5 Suppose u∗ : g End(Vu) is reducible. That means there is a nonzero proper subspace W of Vu that is invariant under u∗(g). Then, owing to Equation 2.2.6, W is invariant under→ ug for g in the image of the exponential map. But this is sufﬁcient to conclude that W is invariant under ug for all g in G because the image of the exponential map generates the connected Lie group G; see [92, Thm. 3.18, p.93].

25 the U(g)-action induced by u∗. So, by Schur’s lemma, the Casimir element, which is in the center of U(g), acts by a scalar on the sum- ∗ mand Vu ⊗ Vu; call that scalar Ω(u). These scalars are the eigenvalues of ∆G on C (G, C)ﬁn. The Peter-Weyl theorem 2.2.22 then 2 implies that there is∞ an orthonormal basis for L (G, C) consisting of smooth eigenfunctions of ∆G, and that the multiplicity of the Ω(u)- 2 eigenfunction is equal to dim(u) . Hence, the heat-trace of ∆G is given by

tr(et∆G ) = tr(et Ω) = dim(u)2et Ω(u). (2.2.26) uX∈Gb In view of the isomorphism 2.2.23, we can obtain the spectrum { Ω(u) } by applying the Laplacian to the irreducible characters u∈Gb χu. Moreover, since χu(e) = dim(u), having a complete informa- tion on the irreducible characters allows us to calculate the heat trace 2.2.26. This becomes a feasible task with H. Weyl’s formula for the irreducible characters, which brings us to the last topic of our review.

2.3 THEWEYLCHARACTERFORMULAAND THESPECTRUMOFTHELAPLACIAN

2.3.1 Throughout this section we assume that the compact Lie group G is connected. We continue to denote the selected bi-invariant metric on G by h , i.

2.3.2 H. Weyl’s approach [96–98] to characterizing the irreducible representations of G starts by considering a connected maximal commutative subgroup T of G. Such a subgroup is known as a maximal torus of G because a compact connected commutative Lie group is isomorphic to a torus (see [18, Prop. 15.3, p. 87]); hence,

dim T T ' U(1). i=1 Y It is a fact that the conjugates of T exhaust all maximal tori of G S −1 and that their union g∈G g Tg is equal to G (see [18, Thm. 16.4, p. 103]). Thus, the dimension of a maximal torus depends only on G; it is called the rank of G. The elements of G that preserve the selected maximal torus T under conjugation constitute the normalizer NG(T) of T relative to G. The elements of NG(T) that act trivially on T under conjugation constitutes T. The quotient

W := NG(T)/T is called the Weyl group of G. If f is a smooth function on T, then an element gT of the Weyl group acts on f by (gT · f)(t) = f(gtg−1) where x in T. A function

26 on T can be extended to a class function on G if and only if it is invariant under the W-action. Thus, the restriction to T gives a linear isomorphism 2 ∼ 2 W Lcl(G, C; µG) − L (T, C; µT ) ,

where µG and µT are the normalized Haar measures on G and T, respectively, and the decoration W→is for the subspace of W-invariants. H. Weyl found that there is some smooth function D on T such that if we modify the measure on T by the factor D/|W| then the restriction map, 2 ∼ 2 D W Lcl(G, C; µG) − L (T, C; |W| µT ) , (2.3.3) is unitary. By carefully studying the image of an irreducible character χ of G under this map, H.→ Weyl obtained a formula for χ in terms of the irreducible characters of T. (We will look into this in more detail in Section 2.3.24.) Bearing in mind that the set of irreducible characters of T can be identiﬁed with the unitary dual Tb of T, a corollary of the character formula is that there is a one-to-one correspondence between Gb and T/Wb . The highest weight theory, which we shall review shortly, gives a means to parametrize T/Wb . As we review some key notions surrounding the character formula, we shall closely follow the global and analytic approach of H. Weyl. There is a corresponding inﬁnitesimal and algebraic approach set forth by E. Cartan [19]; for that, we refer to [88].

2.3.4 WEIGHTS. Recall that any representation of a compact connected Lie group G on a Hilbert space is a direct sum of ﬁnite- dimensional irreducible representations. If G is commutative, so that G is isomorphic to a torus r T = U(1), i=1 Y then the irreducible representations are all 1-dimensional6. Hence, for tori, the following notions are identical: (a) a 1-dimensional representation, (b) an irreducible representation, (c) the character of an irreducible representation. Thus, the unitary dual Tb of T has a natural structure of a commutative group. An irreducible character of an r-dimensional torus must be of the form r θ : T = U(1) C, (2.3.5) i=1 x = (eixY1 , . . . , eixr ) 7 θ(x) = ei(n1x1+···+nrxr), → r where (n1, . . . , nr) ∈ Z . As we have noted earlier, the character θ → 6 Suppose u : T Aut(V) is an irreducible representation. Since T is commutative, u(x): V V is an intertwiner for any x in T. Thus, by Schur’s lemma, each u(x) is a scalar multiple→ of the identity map on V. So any 1-dimensional subspace of V is an invariant→ subspace. Since V is irreducible, V must be 1-dimensional.

27 is just the irreducible representation of T on C where x ∈ T acts on z ∈ C by x · z = θ(x)z. The induced Lie algebra representation of H ∈ t is given by:

d H · z = θ∗(H)z = θ(exp sH)z. (2.3.6) ds 0 r If we identify the Lie algebra t of T with R , then the exponential map takes the form r r exp : t ' R T = U(1), (2.3.7) i=1 (x , . . . , x ) 7 (eix1 ,Y . . . , eixr ). 1 r → Using the maps and equations 2.3.5–2.3.7, we have, for H in t, → θ∗(H) = i(n1x1 + ··· + nrxr), (2.3.8) r where xi is the ith component of the vector in R that corresponds r to H under the identiﬁcation of t with R . We see that θ∗ is a linear function t C whose value is purely imaginary; it is called the complex weight of θ. The corresponding real weight is the linear functional → µ := −iθ∗. In terms of µ, the action of H ∈ t on z ∈ C is given by H · z = iµ(H)z. We will be dealing with real weights most of the time, so when we simply say “weights” it is to be understood as real weights. ∗ We denote by ΛT ⊆ t the set of weights that come from irreducible characters of T; in other words, ΛT is the image of the map Tb t∗, θ 7 −iθ∗. We can conclude from Equation→2.3.8 that → ∗ −1 ΛT = { µ ∈ t | µ(H) ∈ 2πZ for all H ∈ t ∩ exp {e}}.

This is a lattice of rank r := dim(T). If we let

Λe := { H ∈ t | exp(2πH) = e },

then ΛT is the lattice that is dual to Λe in the following sense: ∗ ΛT = { µ ∈ t | µ(H) ∈ Z for all H ∈ Λe }. Moving on to the general case, allow the compact connected Lie group G to be noncommutative. Let u : G Aut(V) be a ﬁnite- dimensional complex representation. Upon restricting u to a maximal torus T in G, the representation space V →is decomposed into 1- dimensional invariant subspaces, each yielding a weight of T. Group- ing the 1-dimensional invariant subspaces according to their weights,

28 we can write V as a direct sum,

V = Vµ1 ⊕ · · · ⊕ Vµ` , (2.3.9) where µk (1 6 k 6 `) are distinct weights. Put in another way, the direct sum 2.3.9 is the eigenspace decomposition under the t-action induced by u; an element H ∈ t acts on v ∈ Vµk as follows

H · v = iµk(H)v.

The linear functionals µ1, . . . , µ` are called the weights of u, and the subspaces Vµk are called the weight spaces of V. Vectors in Vµk are called weight vectors with weight µk. The decomposition 2.3.9 is called the weight space decomposition of V.

2.3.10 We may talk about the weights of a representation u even if it is a real representation, by declaring its weights to be those of the complexiﬁcation of u, that is, the representation on VC = V ⊗ C obtained by extending the action of ug (g ∈ G) by C-linearity.

2.3.11 ROOTS. The most important example of weights coming from of a real representation are the weights of the adjoint representation Ad : G Aut(g). To ﬁnd the weights, we must extend the representation space to the complexiﬁcation gC of g and restrict the domain of the representation→ to a maximal torus T of G. That gives us the weight space decomposition:

gC = gC,0 ⊕ gC,α1 ⊕ · · · ⊕ gC,αk , (2.3.12) where gC,α denotes the weight space corresponding to the weight α; the nonzero weights α1, . . . , αk of the adjoint representation are called the roots of G; the space gC,αi is called the root space associated to the root αi of G, and the decomposition 2.3.12 is called the root space decomposition of gC. We shall denote the set of roots of G by Φ.

Note that the zero weight space gC,0 is the centralizer of tC, that is, the subalgebra of gC that commutes with every element in tC. Because t is a maximal commutative subalgebra of g (otherwise, T cannot be a maximal and commutative subgroup of G), we have

gC,0 = tC. Therefore, the root space decomposition 2.3.12 can be rewritten as

gC = tC ⊕ gC,α1 ⊕ · · · ⊕ gC,αk . (2.3.13) Using the inner product h , i on g, one can ﬁnd, for each linear ∗ functional µ ∈ t , a unique vector Xµ ∈ t such that

µ(H) = hXµ,Hi, ∀H ∈ t. (2.3.14) This gives a one-to-one correspondence between t and t∗; and the

29 inner product h , i can be transferred to t∗ by setting

hµ, νi := hXµ,Xνi. This inner product on t∗ is also invariant under the Adˇ (G)-action (see Section 2.2.5 for the notation). We adopt the notation

Xµ(ν) := ν(Xµ). The set Φ of roots of G satisﬁes the following properties (see [18, Ch. 19 and 23]):

(i) Each root space gC,α is 1-dimensional. (ii) If α ∈ Φ, then λα ∈ Φ if and only if λ = ±1.

(iii) The root spaces gC,α and gC,β are orthogonal if α 6= ±β.

(iv) The subspace [gC,α, gC,−α] is a 1-dimensional subspace of tC. And the subspace gC,α ⊕gC,−α ⊕[gC,α, gC,−α] forms a Lie algebra isomorphic to sl(2, C).

(v) Let Hα be the unique element in [gC,α, gC,−α] such that

α(Hα) = 2.

Then Hα is a vector in t that satisﬁes exp(2πHα) = e. And β(Hα) ∈ Z for all β in Φ. ∗ ∗ (vi) For α in Φ, the linear transformation sα : t t , µ 7 µ − µ(Hα)α, is a reﬂection that sends α to −α. Moreover, sα permutes the roots. → → (vii) The compact connected Lie group G is semisimple if and only if Φ spans t∗.

The vector Hα is called the coroot of α. The reflections sα are called the Weyl reflections; they are precisely the isometries of t∗ that are induced by the conjugation action of the Weyl group W = NG(T)/T on T. According to property (ii) above, the roots come in pairs (α, −α). It is helpful to distinguish one from each pair and call the selected ones “positive”. To that end, divide t∗ into disconnected regions by ∗ excluding from t the hyperplanes hα := ker(1 − sα), α ∈ Φ. So we ∗ S have the set t \ α∈Φ hα, and the Weyl group acts freely on it. A ∗ S connected component of t \ α∈Φ hα is called a Weyl chamber. Pick a Weyl chamber K◦. It is mapped to another Weyl chamber under the W-action; in fact, K◦ is a fundamental domain of the free W-action ∗ S ◦ on t \ α∈Φ hα. Once we have our choice for K , we define the set of positive roots as + ◦ Φ := { α ∈ Φ : hα, νi > 0, ∀v ∈ K }.

30 Note that α ∈ Φ+ implies −α ∈ Φ \ Φ+. So we call the roots in Φ− := Φ \ Φ+ the negative roots. We have a disjoint union: Φ = Φ+ t Φ−. This decomposition of Φ into positive and negative roots depends on which Weyl chamber we select for K◦. From now on, we shall assume that the choice for K◦ has been made. The closure K of K◦ is called the fundamental Weyl chamber of our choice.

Remark. Since roots are special type of weights, they span a sublattice ΛΦ of ΛT , called the root lattice of G. The lattice Λcoroot spanned by the coroots is called the coroot lattice of G; it is a sublattice of Λe.

2.3.15 FUNDAMENTAL WEIGHTS. The coroots, or rather, their dual vectors relative to the inner product can be used to parametrize the lattice ΛT . To that end, choose a special basis for the lattice ΛΦ spanned by the roots. We do this by collecting the positive roots that cannot be written as a sum of other positive roots; such roots are + said to be simple. The set Φs of simple roots is a Z-basis for the root lattice ΛΦ, and an R-basis for ΛΦ ⊗ R. Suppose + Φs = {α1, . . . , α`}. Consider the set of corresponding simple coroots in t:

Σ = {Hα1 ,...,Hα` }. ∗ Let Σ = {λ1, . . . , λ`} be the dual basis for ΛΦ ⊗ R relative to Σ, so that

λi(Hαj ) = δij. The linear functionals in Σ∗ are called the fundamental (dominant) weights of G. They satisfy the following properties (see [18, Ch. 21, Ch. 23; 34, § 3.11]): (i) They lie in the boundary walls of the fundamental Weyl chamber K.

` ∗ (ii) An element µ = i=1 xiλi in t lies in K if and only if xi > 0 for all i. P (iii) The sum of the fundamental weights is equal to half the sum of the positive roots: 1 ρ := λ1 + ··· + λ` = α. 2 + αX∈Φ (iv) The fundamental weights are part of the set ∗ Λg = { µ ∈ t | µ(Hα) ∈ Z for all α ∈ Φ }.

Note that Λg is dual to Λcoroot in the sense that ∗ Λg = { µ ∈ t | µ(H) ∈ Z for all H ∈ Λcoroot }.

31 We have

ΛT ⊆ Λg.

By duality, we have

Λe ⊇ Λcoroot.

(v) If the compact connected Lie group G is semisimple, then the quotient group Λe/Λcoroot is isomorphic to the fundamental group of G:

Λe/Λcoroot ' π1(G).

By duality, Λg/ΛT is isomorphic to the unitary dual of π1(G):

Λg/ΛT ' π\1(G).

We distinguish Λg and ΛT by calling their elements the algebraically integral weights and the analytically integral weights of G, respectively. The weights in Λg ∩ K are said to be dominant; the ones in ◦ Λg ∩ K are said to be strictly dominant. Note that ◦ Λg ∩ K = ρ + (Λg ∩ K). (2.3.16)

2.3.17 PARTIAL ORDERING ON THE WEIGHT LATTICE. We have put some effort in reviewing the notions surrounding the roots of G. This is because they can be used to give a partial ordering on the weight lattice ΛT and, thus, allow us to talk about “highest weights” in a finite-dimensional representation of G. To wit, for µ and λ in ΛT , we say that λ is higher than µ and write µ . λ if and only if λ can be written as λ = µ + nαα + αX∈Φ for some nonnegative integers nα. That this defines a transitive relation on ΛT is immediate. That this is a reflexive relation is equiv- + alent to saying that if α∈Φ+ cαα = 0 with cα > 0 for all α ∈ Φ + then cα = 0 for all α ∈ Φ ; for a proof that this is the case, see P [34, p. 147]. In short, we do have a partial ordering on ΛT . Now suppose we have a finite-dimensional representation u of G. Let A = {µ1, . . . , µk} be the set of weights of this representation. We say that µi ∈ A is a highest weight of this representation if it is a maximal element of A with respect to the partial ordering ..

2.3.18 WEYL INTEGRATION FORMULA. Since the characters of G are class functions, their restrictions to T are W-invariant; hence, we have a linear map 2 2 W Lcl(G, C; µG) L (T, C; µT ) , where µG and µT are any Haar measures on G and T, respectively. A W-invariant function on T extends→ to a class function on G, so the

32 above map is a linear isomorphism. H. Weyl found out that a slight modification of the measure on T gives a unitary map 2 ∼ 2 W Lcl(G, C; µG) − L (T, C; cDµT ) , where c is the constant µG(G)/(µT (T)|W|), and D is the function on T defined by → D(x) = (eα(H)/2 − e−α(H)/2) + αY∈Φ where H ∈ t ∩ exp−1{x}. This is a consequence of the celebrated Weyl Integration Formula: 1 µ (G) h i f(g) µ (g) = G f(gxg−1) µ (g¯) |D(x)|2 µ (x), G |W| µ (T) G/T T ZG T ZT ZG/T (2.3.19) which holds for any integrable function f on G; hereg ¯ denotes the image of g under the quotient map π : G G/T and µG/T is the −1 quotient measure on G/T defined by µG/T (U) = µG(π (U)) for any open set U in G/T. For a proof of the integration→ formula, see [34, Thm. 3.14.1, p. 185]. If f is a class function, then the integration formula simplifies to:

1 µG(G) f(g) µ (g) = f(t) |D(t)|2 µ (t). (2.3.20) G |W| µ (T) T ZG T ZT There is a corresponding formula for integration over g. Let ωG and ωT be the invariant volume forms on G and T, respectively, such that G ωG = µG(G) and T ωT = µT (T). The values of ωG and ωT at the identity determines a volume form for g and t, respectively. R R Denote the Lebesgue measure they deﬁne on g and t by dX and dH, respectively. Then, for any integrable function φ on g, 1 µ (G) φ(X) dX = G φ(Ad H) µ (g¯) |δ(H)|2 dH, |W| µ (T) g G/T Zg T Zt ZG/T (2.3.21) where δ(H) = α(H). (2.3.22) + αY∈Φ See [34, Thm. 3.14.1, p. 185] for a proof. If φ is Ad(G)-invariant, then 1 φ(X) dX = φ(H) |δ(H)|2 dH. (2.3.23) |W| Zg Zt

2.3.24 WEYL CHARACTER FORMULA. Notice that the function D is antisymmetric with respect to the W-action. Meanwhile, the restriction of a class function f on G to T is W-invariant. So the product of D and the restriction fT gives a W-antisymmetric function. This gives us a vector space isomorphism 2 ∼ 2 1 −W L (G, C; µG) − L (T, C; µT ) , cl |W| (2.3.25) f 7 fT D, → → 33 2 1 −W 2 1 where L (T, C; |W| µT ) denotes the subspace of L (T, C; |W| µT ) that consists of the W-antisymmetric elements. If we assume that µG and µT are the normalized Haar measures, then the integral formula 2.3.20 implies that the map 2.3.25 is unitary. Now, owing to the isomorphism 2.2.23, the set

{ χu | u ∈ Gb } 2 of irreducible characters of G is an orthogonal basis for Lcl(G, C; µG). It is, in fact, an orthonormal basis (see [18, Thm. 2.4, p. 12]). This basis yields, through the unitary map 2.3.25, an orthonormal basis 2 1 −W for L (T, C; |W| µT ) . H. Weyl found that the orthonormal basis for 2 1 −W L (T, C; |W| ωT ) thus obtained is

◦ det(w) θw·λ λ ∈ ΛT ∩ K ,

w∈W X where θλ denotes the irreducible character of T deﬁned by θλ(x) = iλ(H) −1 ◦ e (H ∈ t∩exp {x}). Note that the W-orbits of weights in ΛT ∩K are free and that these orbits exhaust all free orbits in ΛT . Hence, ◦ for each character χu of u in Gb, there is a unique weight λ in ΛT ∩K such that χuT D = det(w) θw·λ, (2.3.26) w∈W X and vice versa. More precisely, H. Weyl found that u and λ in the above equation are related by: λ = µ + ρ, (2.3.27) where µ is a highest weight of u and ρ is the sum of the fundamental dominant weights. The uniqueness of λ determined by u implies the uniqueness of the highest weight µ; moreover, by Equation 2.3.16, the highest weight µ lies in ΛT ∩ K. In short, we have a one-to-one correspondence:

Gb ΛT ∩ K, (2.3.28) u 7 highest weight µ of u. For this reason, the representation↔ space of an irreducible representation of G that has highest→ weight µ is often called a highest weight module with highest weight µ. We point out again that the one-to-one correspondence 2.3.28 is a consequence of Equations 2.3.26 and 2.3.27 which are collectively referred to as the Weyl Character Formula. They are usually combined and presented as Altµ+ρ χ = (2.3.29) uT D where Altλ := det(w) θw·λ. w∈W X

34 Applying the Weyl Character Formula to the trivial representation, whose highest weight and character are µ = 0 and χµ ≡ 1, we get D = Altρ. (2.3.30) This is known as the Weyl Denominator Formula. Inserting the expression 2.3.30 for D back into the character formula 2.3.29 and taking the limit limx e χuT (x) yields the Weyl Di- mension Formula for the dimension of the highest weight module V(µ) with highest weight µ: → hα, µ + ρi dim V(µ) = . (2.3.31) + hα, ρi αY∈Φ Remark. The signiﬁcance of the weight ρ is that it is the highest weight of the representation of T on the spinor space S of Cl(p), where p is the orthogonal complement of t in g; the action of T on S is by the group homomorphism T Spin(p) that is the lift of the homomorphism T SO(p) given by the adjoint action of T on p. (We discuss this representation in detail→ on pages 126–127.) → Spin(p) :

T / SO(p) Ad The ubiquitous appearance of the weight ρ in the representation theory of G may be taken as a suggestion for using Dirac operators instead of the Casimir element in characterizing the irreducible representations of G.

2.3.32 It is worthwhile to give R. Bott’s description [16] of the Weyl Character Formula using the representation ring. Let [V] denote the equivalence class of ﬁnite-dimensional G-vector spaces that are G-isomorphic to V. Let R(G)+ denote the set of such equivalence classes modulo the relation [V] + [W] = [V ⊕ W], (2.3.33) [V][W] = [V ⊗ W]. Then representation ring R(G) of G is the abelian group generated by + R(G) . As a Z-module, we have M R(G) = Z · [Vu], u∈Gb where Vu denotes the representation space of u. If we identify [V] with the character χV of the representation on V, then the relations in 2.3.33 are true equations of characters. For this reason, R(G) is also known as the character ring; in this point of view, the elements of R(G) are called virtual characters.

35 The character ring is functorial; if ι : K G is a homomorphism of compact Lie groups, then the composition of a representation of G with ι induces a ring homomorphism → ι∗ : R(G) R(K). If ι is the inclusion map of a Lie subgroup, then ι∗ is just the restriction map. → Now consider the representation ring of a maximal torus T of G. The action of the Weyl group W on T induces a W-action on R(T). The isomorphism 2.3.25 can then be stated as follows [16, Thm. A, p. 175]: Let ι : T , G be the inclusion map of a maximal torus T of G. The induced homomorphism ι∗ : R(G) R(T) yields a ring isomorphism → ∼ R(G) −− R(T)W, → ι∗ W where R(T) denotes the ring of→W-invariants in R(T). An element x of R(T) is said to be W-alternating if w · x = sgn(w)x. Denote by R(T)−W the subspace of R(T) that consists of the W-alternating elements. Of particular interest is the W-alternating element [Λ] := [Cρ] (1 − [C−α]), + αY∈Φ where [Cµ] denotes the irreducible T-vector space with weight µ. The Weyl Character Formula can then be restated as follows [16, Thm. D, p. 178, Cor. 6.1, p. 178]: Suppose π1(G) has no 2-torsion. (a) The alternating element [Λ] generates R(T)−W as a free module over R(T)W.

(b) If V is an irreducible G-vector space. Then ∗ [Λ]ι [V] = sgn(w)[Cw·µ] (2.3.34) w∈W X for some weight µ in ΛT . Conversely, for any weight µ in ΛT , there is some irreducible G-vector space V such that Equa- tion 2.3.34 holds up to sign.

Remark. There is a well-known construction for inducing a representation of G from a representation of a closed Lie subgroup K of G. Suppose E is a ﬁnite-dimensional complex inner product space on which K acts as unitary transformations. Consider the right K-action on G × E given by (g, v) · k = (gk, k−1 · v). The orbit space

E(G) := G ×K E is a vector bundle over G/K with fibers isomorphic to E (see [63, Prop. 5.4, p. 55]). For each point x in the fiber overg ¯ := gK (g ∈ G), let kxkE denote the norm of the vector in E that corresponds to x under the identification of the fiber overg ¯ with E. Define the L2-

36 norm of a section σ of E(G) by

2 2 kσkL2 = kσ(g¯)kE dg,¯ ZG/K where dg¯ is the quotient measure on G/K relative to the normalized Haar measures on G and K. Let Γ 2E(G) be the L2-closure of the space of smooth sections of E(G). The left-regular action of G on Γ 2E(G) makes it a G-vector space. (It is usually inﬁnite-dimensional.) This representation is called the induced representation of G associated to the representation of K on E. This induction gives rise to a group homomorphism

ι : R(K) R(G) := · [V ]. ∗ b Z u (2.3.35) uY∈Gb → R. Bott calls the codomain Rb(G) as the formal representation group of G and the homomorphism 2.3.35 as the formal induction. The group Rb(G) contains the ring R(G), but the multiplication in R(G) does not extend to a well-deﬁned multiplication in Rb(G).

2.3.36 EIGENVALUES OF THE CASIMIR (LAPLACIAN). Owing to the one-to-one correspondence 2.3.28, we may use ΛT ∩K to parametrize Gb and write the isomorphism 2.2.21 as M ∗ C (G, C)ﬁn ' V(µ) ⊗ V(µ), (2.3.37) µ∈Λ ∩K ∞ T where V(µ) is an irreducible G-representation space with highest weight µ. Each summand V(µ)∗ ⊗ V(µ) is an eigenspace of the Casimir Ω (see Section 2.2.24). Let Ω(µ) denote the corresponding eigenvalue. One of the motives for reviewing the Weyl formulas was that the the eigenvalue Ω(µ) is by which the Laplacian acts on the character of the representation on V(µ) (see the end of Sec- tion 2.2.24).

2.3.38 THEOREM. Let G be a compact, simply connected, semisimple Lie group equipped with a bi-invariant metric. The scalar by which the Casimir element Ω acts on the irreducible representation space of G with highest weight µ is Ω(µ) = −kµ + ρk2 + kρk2, where ρ is half the sum of the positive roots and k · k is the norm on g∗ induced by the metric.

Proof. The Weyl Character Formula 2.3.29 is an explicit expression for the restriction of χµ to T. So we need to know the differential operator on T whose action on the restriction χµT is equal to ∆Gχµ. If G is compact, simply connected, and semisimple, then it is known

37 that, for any smooth class function f on G, 1 (∆ f) = ◦ ∆ ◦ D + kρk2 f G T D T T where ∆T is the Laplacian on T (see [53, pp. 10 and 19]). The compositions appearing in the above equation are to be understood as that of differential operators; for instance, D represents the multiplication operator by the function D. Substituting f with χµ, we have 1 Ω(µ) χ = ◦ ∆ ◦ D + kρk2 χ . µT D T µT Applying the Weyl Character Formula 2.3.29, we get 2 Ω(µ)Altµ+ρ = ∆T Altµ+ρ + kρk Altµ+ρ. (2.3.39)

To calculate ∆T Altµ+ρ, we need to know how ∆T acts on an irreducible character θλ of T with weight λ. For any x in T, we have ihX ,Hi θλ(x) = e λ where Xλ is deﬁned by Equation 2.3.14 and H ∈ t ∩ exp−1{x}. Hence,

∆T θλ = −kXλkθλ = −kλkθλ. Therefore, 2 ∆T Altµ+ρ = −kµ + ρk Altµ+ρ. Inserting this into Equation 2.3.39 proves the theorem.

Remark. There is also a Lie algebraic proof; see [62, Prop. 5.28, p. 295].

38 THEVOLUMEOF3 ACOMPACTLIEGROUP

S a first application of the asymptotic expansion of the heat trace, A we shall verify Harish-Chandra’s formula [51, p. 203, Lem. 4] for vol(G)/ vol(T), the ratio of the volume of a compact connected Lie group G to that of a maximal torus T in G, where each volume is measured with respect to the bi-invariant metric generated by an Ad(G)-invariant inner product h , i on the Lie algebra g of G. The significance of this ratio is its appearance in the Weyl Integration Formula 2.3.19. Harish-Chandra’s formula states that vol(G) = 2πhα, ρi−1. (3.0.1) vol(T) + αY∈Φ Here Φ+ is the set of (selected) positive roots, ρ is half the sum of the positive roots, and h , i is the inner product on the dual space t∗ of the Lie algebra of T induced by the inner product on g. An interesting case is when the compact Lie group G is semisimple. Then we may take the negative of the Killing form 2.1.4 as the Ad(G)-invariant inner product on g. Then the right-hand side of Equation 3.0.1 depends only on g. For the rest of this chapter, we shall assume that the compact connected Lie group G is semisimple and simply connected; there is no loss of generality, as far as vol(G)/ vol(T) is concerned, for the following reasons. By the general theory of compact connected Lie groups, every compact connected Lie group is isomorphic to G ' (R × S)/F (3.0.2) where R is a torus, S is a compact, connected, simply connected, semisimple Lie group, and F is a finite abelian subgroup of R × S (see [62, Thm. 4.29, p. 250]). So F is contained in a maximal torus

39 Te of R × S. Then T/Fe is a maximal torus of (R × S)/F; let T be the corresponding maximal torus in G under the isomorphism 3.0.2. The Lie groups G, R × S, and (R × S)/F all have isomorphic Lie algebras. So the Ad(G)-invariant inner product on g induces a bi-invariant metric on the Lie groups G, R × S, (R × S)/F, and their respective maximal tori. Their Riemannian volumes satisfy vol(R × S) vol((R × S)/F) vol(G) = = . vol(Te) vol(T/Fe ) vol(T) Hence, we may assume that F is trivial so that G ' R × S. Now the maximal torus Te of R×S is of the form R×TS, where TS is a maximal torus of S. Hence, vol(R × S) vol(R × S) vol(S) = = . vol(Te) vol(R × TS) vol(TS) This shows that we may as well assume that G = S, that is, G is semisimple and simply connected. Derivations of the formula 3.0.1 can also be found in [34, 38, 40, 72]; among these, H. D. Fegan [38] uses the heat trace approach. Our calculation differs from H. D. Fegan’s, in that we use the Euler- Maclaurin Formula and the Weyl Integration Formula as our main tools for calculation.

3.1 THEEULER-MACLAURINFORMULA

Let f be a smooth function on the real line. The Euler-Maclaurin For- n n mula [36, 37, 73] relates a sum x=0 f(x) to the integral 0 f(x) dx. We are interested in this formula since the heat trace is a sum of the n P R type x=0 f(x) but with n = . The formula states, for N ∈ N, n P n N B f(x) = f(x) dx + (−∞1)q q f(q−1)(n) − f(q−1)(0) + R , q! N x=0 0 q=1 X Z X (3.1.1) where the coefﬁcients Bq are the Bernoulli numbers deﬁned by the power series

x Bq : q q td(x) = −x = (−1) x , 1 − e ∞ q! q=0 X and RN is the remainder term, which can be estimated by n 2ζ(N) (N) |RN| f (x) dx. (3.1.2) 6 (2π)N Z0 Here ζ is the Riemann zeta function. Suppose k=0 f(k) exists and (q) limx f (x) = 0 for all q ∈ N. Then we may take n = in the P∞ →∞ ∞

40 formula 3.1.1, which gives us N q Bq (q−1) f(k) = f(x) dx + (−1) f (0) + RN. (3.1.3) ∞ q! k=0 0∞ q=1 X Z X If furthermore RN 0 as N (for instance, when f is a polynomial), we have → → ∞∂ f(k) = td f(x) dx. (3.1.4) ∞ ∂h h=0 k=0 −∞h X Z (This expression for the Euler-Maclaurin Formula ﬁrst appeared in [81].) The following Lemma can be proved using the Euler-Maclaurin Formula. We shall make use of it later on.

3.1.5 LEMMA. Let A and B be real numbers with A > 0, and let m be any nonnegative integer. Let 2m −t(Ax2+Bx) ft(x) = x e . Consider the sum, for t > 0,

S(t) = ft(x). ∞ x=0 X Then, m+1/2 1/2 t S(t) = ft(x) dx + O(t ) Z0∞ for t 0+.

Proof. By Equation 3.1.3 with N = 1, →

S(t) = ft(x) dx − B1ft(0) + R1. Z0∞ By the estimate 3.1.2, the absolute value of R1 is bounded by, up to a constant factor,

2 2 2m−1 −t(Ax +Bx) 2m+1 2m −t(Ax +Bx) 2mx e − t(2Ax + Bx )e dx. Z0∞ By a change of variable of the type x 7 αx + β, α > 0, the above integral can be recast in the following form:

2 → 2 −tx −tx P1(x)e + tP2(x)e dx, Zα∞ where P1(x) and P2(x) are polynomials of degree 2m − 1 and 2m + 1, respectively. This integral is bounded by, up to a constant factor,

−tx2 −tx2 |P1(x)|e dx + t |P2(x)|e dx. (3.1.6) Z−∞ Z−∞

∞ ∞

41 It is known that, for a nonnegative integer n,

2 1 n + 1 |xn|e−tx dx = Γ , (3.1.7) t(n+1)/2 2 Z−∞ where Γ denotes the Gamma function. Hence, we see that the in- ∞ tegral 3.1.6 is bounded by a quantity that is of O(t−m) for t 0+. −m−1/2 Meanwhile, the integral 0 ft(x) dx is of O(t ). Hence, ∞ → m+1/2 R 1/2 t S(t) = ft(x) dx + O(t ). Z0∞ 3.2 ANEXAMPLE:SU(2)

3.2.1 Before we launch into general arguments, we take SU(2) as a concrete example to illustrate our strategy.

3.2.2 SETUP. Let G be SU(2), the set of 2 × 2 unitary matrices of determinant 1. Its Lie algebra g is su(2), the set of 2 × 2 complex matrices that are traceless and antihermitian. It is known that G is diffeomorphic to the 3-sphere (see [34, § 1.2.B]); hence, it is compact, connected, and simply connected. As we shall shortly see, the Killing from κ on g is negative deﬁnite; so G is semisimple. For the maximal torus T in G, we take T = eit 0 : t ∈ ' U(1). 0 e−it R We pick the following matrices as the basis vectors for g: 0 i 0 1 i 0 X = i 0 ,Y = −1 0 ,H = 0 −i . The vector H lies in the Lie algebra t of T. The matrices [adX], [adY], and [adH] for the adjoint representation of the basis vectors X, Y, and H, respectively, are given by ··· ·· −2 · 2 · [adX] = ·· 2 , [adY] = ··· , [adH] = −2 ·· . · −2 · 2 ·· ··· Here the dots represent 0. From this, we can calculate the matrix [η] for the bilinear form η = −κ on g; it turns out that 8 ·· [η] = · 8 · . (3.2.3) ·· 8 We take the bi-invariant metric induced by η as the metric for G.

3.2.4 ROOTS. The complexiﬁcation of su(2) is sl(2, C). The standard basis for sl(2, C) is given by the following matrices: 1 J+ = 0 1 = (−iX + Y), 0 0 2 1 J− = 0 0 = (−iX − Y), 1 0 2 0 1 0 J = 0 −1 = −iH.

42 The basis elements satisfy the Lie bracket relations [J+,J−] = J0, (3.2.5) [J0,J±] = ±2J±. (3.2.6) Equation 3.2.6 shows that there are two complex roots ±µ, which are characterized by ±µ(H) = ±iµ(J0) = ±2i. Denoting the corresponding real roots as ±α, we have ±α(H) = ±2. The Weyl group is the cyclic group of order 2 generated by σ : α 7 −α. If we take α as the basis for the dual space t∗ of t, thereby ∗ identifying t with the real line R, then σ is the reﬂection with respect→ to the origin. We choose the closed half-line [0, [ as the fundamental Weyl chamber K. This amounts to choosing α as the positive root. ∞

3.2.7 DOMINANT WEIGHTS. The dual vector Xα of α is characterized by α(H) = hXα,Hi = 2. (3.2.8)

This implies Xα = H/4. Since α(Xα) = hXα,Xαi = 1/2, we have 1 hα, αi = hX ,X i = . (3.2.9) α α 2 There is only one fundamental dominant weight, which is 1 ρ := α. 2 We have 1 1 hρ, ρi = hα, αi = . (3.2.10) 4 8

Since SU(2) is simply connected, the weight lattice ΛT is spanned by ρ (see Section 2.3.15). The set of dominant weights is

ΛT ∩ K = { `ρ | ` = 0, 1, 2, . . . }.

3.2.11 HEAT TRACE. Let V(`) denote the highest weight module with highest weight `ρ. Then, by the isomorphism 2.3.37,

M ∗ C (G, C)ﬁn ' V(`) ⊗ V(`) . ∞ `=0 ∞ By the Weyl Dimension Formula 2.3.31, hα, (` + 1)ρi dim V(`) = = ` + 1. (3.2.12) hα, ρi

43 The value of the Casimir on V(`) is (Theorem 2.3.38): (` + 1)2 1 Ω(`) = −(` + 1)2kρk2 + kρk2 = − + . (3.2.13) 8 8 Therefore, the heat trace is

2 tret∆G = dim V(`)2 et Ω(`) = et/8 (` + 1)2 e−t(`+1) /8. ∞ ∞ `=0 `=0 X X (3.2.14) For the volume of G, we are only interested in the leading term of the asymptotic expansion of the heat trace. Thus, we only need to calculate 2 −t(`+1)2/8 Z(t) := (` + 1) e . ∞ `=0 X Deﬁne the polynomial d(`) in ` by d(`) = `. Then,

2 Z(t) = d(n)2e−tn /8. ∞ n=0 X

3.2.15 ASYMPTOTIC EXPANSION. Set

2 −tx2/8 ft(x) := x e . Then

Z(t) = ft(x). ∞ x=0 X By the Euler-Maclaurin Formula 3.1.3, N 1 Bp+1 (p) Z(t) = f (x) dx + f(0) − f (0) + R . (3.2.16) t 2 (p + 1)! t N 0∞ p=1 Z X We are only interested in the leading term of the asymptotic expansion of Z(t). By Lemma 3.1.5,

3/2 3/2 t Z(t) = t ft(x) dx + O(t) (3.2.17) Z0∞ for t 0+.

3.2.18→SYMMETRY. The symmetry under the Weyl reﬂection now comes into play. The function ft(x) is invariant under the reﬂection x 7 −x. So 1 Z(t) = f (x). 2 t → x∈ XZ

44 And Equation 3.2.17 can be rewritten as t3/2Z(t) = t3/2I(t) + O(t), (3.2.19) where 1 I(t) := f (x) dx. (3.2.20) 2 t Z−∞ This integral is essentially a Gaussian integral, which can be easily ∞ computed (See Equation 3.1.7): √ 4 2π I(t) = . t3/2 So Equation 3.2.19 gives √ t3/2Z(t) = 4 2π + O(t). (3.2.21)

3.2.22 VOLUMEOF G AND T . Owing to Weyl’s law 1.0.6, the asymptotic equality 3.2.21 yields √ vol(G) = 32 2π2. (3.2.23) The volume of T is given by 2π 1/2 1/2 vol(T) = hHα,Hαi dt = 2πhHα,Hαi , Z0 where Hα is the coroot in t associated to α. (We shall prove this in Lemma 3.3.9(b).) The coroot Hα is characterized by 1 λ(H ) = α(H ) = 1. α 2 α

Thus, Hα = H. As the matrix 3.2.3 shows, hH, Hi = 8. So the volume of T is √ vol(T) = 4 2π. Combining with Equation 3.2.23, we have vol(G) = 8π. (3.2.24) vol(T) This agrees with Harish-Chandra’s formula 3.0.1, which gives vol(G) = 2πhα, α/2i−1 = 4πhα, αi−1 = 8π, vol(T) where we have used Equation 3.2.9 in the last step.

3.3 THEGENERALCASE

3.3.1 We now consider the general case. Our aim is to prove Equa- tion 3.3.16 for any compact connected Lie group G. We shall follow in outline the calculation presented in Section 3.2. The only new ingredient that will appear is the Weyl Integration Formula. We adopt the following notation for the dimension of G, T, and

45 G/T:

n := dim G, r := dim T, 2m := dim G − dim T = n − r. We can read from the root space decomposition 2.3.13 that 2m is equal to the number of roots of G. Since half of the roots are positive (see page 31), we have m = |Φ+|. (3.3.2)

3.3.3 As we explained in the beginning of this chapter, we may assume that G is compact, connected, simply connected, and semisimple. In this case, there is a one-to-one correspondence between the unitary dual Gb of G and the set ΛT ∩K of dominant weights of G. For each λ ∈ ΛT ∩ K, let V(λ) denote a highest weight module with highest weight λ. By Equation 2.2.26 and Theorem 2.3.38, we have, for the heat trace, tr(et∆G ) = dim V(λ)2et Ω(λ), λ∈Λ ∩K XT where Ω(λ) = −kλ + ρk2 + kρk2. The dimension of V(λ) can be calculated by the Weyl Dimension Formula 2.3.31. If we let

α∈Φ+ hα, λi d(λ) := , (3.3.4) + hα, ρi Qα∈Φ then the Weyl Dimension FormulaQ takes the form dim V(λ) = d(λ + ρ). So the heat trace can be expressed as

2 2 tr(et∆G ) = etkρk d(λ + ρ)2e−tkλ+ρk . (3.3.5) λ∈Λ ∩K XT The leading term of the asymptotic expansion of the heat trace 3.3.5 is identical with that of 2 −tkλ+ρk2 2 −tkλk2 Z(t) := d(λ + ρ) e = d(λ) e . λ∈Λ ∩K XT λ∈(ΛXT ∩K)+ρ (3.3.6)

Note that the shifted index set (ΛT ∩ K) + ρ is the set of the weights that lie in the interior of the Weyl chamber. (This can be deduced from the properties of the fundamental weights that we listed in Section 2.3.15.) Hence,

2 Z(t) = d(λ)2e−tkλk . (3.3.7) λ∈Λ ∩K◦ XT But since the restriction of the function d to the boundary of the ◦ Weyl chamber is zero, we may replace the index set ΛT ∩ K of the

46 sum 3.3.7 with ΛT ∩ K and write 2 Z(t) = d(λ)2e−tkλk . (3.3.8) λ∈Λ ∩K XT ∗ 3.3.9 LEMMA. Let µt∗ denote the Lebesgue measure on t induced by the inner product h , i. Let vol(P) be the volume of the fundamental parallelepiped formed by the fundamental weights of G in t∗.

(a) For t 0+, we have tn/2Z(t) = tn/2I(t) + O(t), (3.3.10) → where d µt∗ (λ) I(t) = d(λ)2e−tkλk . (3.3.11) vol(P) ZK (b) The volume of the maximal torus is related to vol(P) by vol(T) = (2π)r vol(P)−1. (3.3.12)

Proof. (a) Denote the set of fundamental dominant weights by

{ λ1, . . . , λr }. ∗ This is a basis for t . Let us use (x1, . . . , xr) to denote the components of a vector in t∗ with respect to this basis. Then Z(t) is of the form

2 −tq(x1,...,xr) Z(t) = ··· d(x1, . . . , xr) e , ∞ ∞ x =0 x =0 X1 Xr where d and q are homogeneous polynomials of degree m and 2, respectively. In terms for the variables (x1, . . . , xr), the integral 3.3.11 takes the form

2 −tq(x1,...,xr) I(t) = ··· d(x1, . . . , xr) e dx1 ··· dxr. Z0∞ Z0∞ Applying Lemma 3.1.5 iteratively to Z(t) proves the assertion. (b) Because G is simply connected, the coroot lattice Λcoroot co- −1 incides with the lattice Λe := t ∩ exp {e}. By duality, the lattice Λg of algebraically integral weights coincides with the lattice ΛT of an- r alytically integral weights. Thus, the simple coroots { Hαi }i=1 form r a basis for Λe, and the corresponding fundamental weights { λi }i=1 form a basis for ΛT . Let Q be the fundamental parallelepiped in t formed by the simple coroots. Let µt be the Lebesgue measure on t induced by the inner product h , i. Since exp(2πHα) = e for any root α, we have vol(T) = (2π)r vol(Q). (3.3.13)

Meanwhile, because the lattice Λe and ΛT are dual to each other, we have vol(P) = vol(Q)−1. (3.3.14) This proves the assertion.

47 3.3.15 THEOREM. Let G be a compact connected Lie group and T its maximal torus. Let g and t be their Lie algebras, respectively. Let h , i denote an Ad(G)-invariant inner product on g and also the induced inner product on the dual space g∗. The Riemannian volume of G and T with respect to the bi-invariant metric generated by h , i are related by vol(G) = 2πhα, ρi−1, (3.3.16) vol(T) + αY∈Φ + 1 where Φ is the set of the selected positive roots of G and ρ = 2 α. α∈Φ+ P Proof. We point out once again that G may be assumed to be simply connected and semisimple (as explained on pages 39–40). The set of roots and the inner product h , i on t∗ are preserved under the action of the Weyl group W. This implies that |d(λ)| and kλk that appear in the integral 3.3.11 are preserved under the W-action; so it is possible to extend the domain of integration to all of t∗ as follows:

1 2 µ ∗ (λ) I(t) = d(λ)2e−tkλk t |W| ∗ vol(P) Zt 1 vol(T) 2 −tkλk2 ∗ = r d(λ) e µt (λ). (3.3.17) |W| (2π) ∗ Zt We have used Equation 3.3.13 for the last equality. We wish to change the domain of integration from t∗ to t via the isomorphism t∗ t, λ 7 Xλ, → where Xλ is deﬁned by the equation → λ(H) = hXλ,Hi, ∀H ∈ t. This isomorphism is precisely through which the inner product on t was transferred to t∗; in particular, the Jacobian determinant of this isomorphism is 1. Next, we have 2 2 2 α∈Φ+ hα, λi α∈Φ+ α(Xλ) d(λ) = 2 = 2 , (3.3.18) + hα, ρi + hα, ρi Qα∈Φ Q α∈Φ and Q 2 Q 2 e−tkλk = e−tkXλk . Thus,

1 vol(T) 2 I(t) = hα, ρi−2 α(X)2 e−tkXk µ (X), |W| (2π)r t α∈Φ+ Zt α∈Φ+ Y Y (3.3.19) where µt is the Lebesgue measure on t induced by the inner product. Recall the Weyl Integration Formula 2.3.23, which states that, for

48 any ad(g)-invariant function f on g,

1 vol(G) 2 f(X) µg(X) = f(X) α(X) µt(X). (3.3.20) g |W| vol(T) t + Z Z αY∈Φ The measure µg on g is again the Lebesgue measure induced by the 2 inner product. Inserting f(X) = e−tkXk into Equation 3.3.20 and rearranging the terms, we get

−tkXk2 2 vol(T) e α(X) µt(X) = |W| f(X) µg(X). t + vol(G) g Z αY∈Φ Z Applying this equation to the right-hand side of Equation 3.3.19, we get 2 vol(T) −2 −tkXk2 I(t) = r hα, ρi e µg(X). (3.3.21) (2π) vol(G) + g αY∈Φ Z The last integral is just a Gaussian integral:

2 πn/2 e−tkXk µ (X) = . g t Zg Hence, 2 n/2 vol(T) −2π I(t) = r hα, ρi . (3.3.22) (2π) vol(G) + t αY∈Φ Inserting this into Equation 3.3.10 and invoking Weyl’s law 1.0.6, we get 2 vol(G) 2m −2 2 = (2π) hα, ρi . vol(T) + αY∈Φ The inner product between two positive roots is nonnegative. (This owes to the deﬁnition of Φ+; see page 30.) Hence, vol(G) = (2π)m hα, ρi−1. vol(T) + αY∈Φ By Equation 3.3.2, we have vol(G) = 2πhα, ρi−1. vol(T) + αY∈Φ

49 THEHEATKERNELOF4 THELAPLACIANON ACOMPACTLIEGROUP

E now come to one of our main objectives — the calculation of W the asymptotic expansion for the heat kernel of the Laplacian ∆G on a compact connected Lie group G that is endowed with a bi-invariant metric h , i. Our aim is to show that one can quickly obtain the asymptotic expansion using Lie algebra methods. The key ingredient is the Duflo isomorphism [32] Duf : S(g)g Z(g), which is an algebra homomorphism from the ad(g)-invariant part of the symmetric algebra of g (elements→ of ad(g) act as inner derivations) to the center of the universal enveloping algebra of g. The domain S(g)g can be identified with the space of constant coefficient differential operators on g that are invariant under the Ad(G)-action, and the image Z(g) can be identified with the space of bi-invariant g differential operators on G. In S(g) is the Laplacian ∆g of the Eu- clidean space g, and in Z(g) is the Laplacian ∆G of the curved space G. The main idea is this: Find a relation between Duf(∆g) and ∆G, and use that to deduce the asymptotic expansion for the heat kernel of ∆G. What makes this a plausible route? Assume for the moment that the relationship between Duf(∆g) and ∆G is simple enough so that the heat kernel of ∆G is a minor modification of the heat kernel of Duf(∆g). The heat kernel of Duf(∆g) is the integral kernel of the operator et Duf(∆g). Recall that the Duflo isomorphism is an algebra isomorphism. So, heuristically, we expect et Duf(∆g) to be equal to “Duf(et∆g )”. Now the integral kernel of et∆g is well-known to be the Gaussian kernel. So our expectation is that, by understanding

50 the geometric meaning of the Duflo isomorphism, we would be able to figure out its effect on the Gaussian kernel and deduce from it the integral kernel of “Duf(et∆g )”, or rather, for et Duf(∆g), which is essentially et∆G . This last statement can be made more precise using the Kashiwara-Vergne conjecture [60] (now proved), which extends the Duflo isomorphism to incorporate invariant distributions with compact support. In this dissertation, however, we shall rely on more elementary means.

Remark. When we speak of a differential operator on a manifold M, we shall assume that its domain is C (M), unless explicitly stated otherwise. ∞

4.1 THEDUFLOISOMORPHISM

4.1.1 INVARIANT DIFFERENTIAL OPERATORS. Let A be any Lie group. Suppose A acts smoothly on a manifold M on the left. Let g be an element of A. For a smooth function f on M, we define the smooth function g · f on M by (g · f)(x) = f(g−1 · x). (4.1.2) For a differential operator D on M, we define the differential operator g · D on M by (g · D)f = g · (D(g−1 · f)). (4.1.3) The differential operator D is said to be invariant if (g·D) = D holds for all g in A. Now let M = A, and consider the action of A on itself by left translations. In this case, the invariant differential operators on A are said to be left-invariant. Let D(A) be the space the left-invariant differential operators on A. Let a be the Lie algebra of A. The construction of the universal enveloping algebra U(a) of a and its identification with D(A), which we reviewed in Section 2.2.10, still applies here, and we have an algebra isomorphism τ : U(a) D(A), (4.1.4) X1 ⊗ · · · ⊗ Xk 7 Xe1 ◦ · · · ◦ Xek. → Here Xe denotes the left-invariant vector field on A generated by X in → a. Suppose D is an invariant differential operator on A. Let Dg denote the value of D at g in A; in other words, for a smooth function f on A, Dgf = (Df)(g). Since Equation 4.1.3 holds for D, we have g−1 · (Df) = D(g−1 · f).

51 Evaluating both sides of this equation at the identity, we get (Df)(g) = (D(g · f))(e). Put in another way, −1 Dgf = De(g · f). In short, D is completely determined by its value at the identity. Now De : C (A) R is a distribution on A that is supported on {e}. So we have∞ a linear map from the space of left-invariant differential operators on→ A to the space of distributions on A with support {e}; ε : D(A) E0 (A), e (4.1.5) D 7 De. This is, in fact, an algebra isomorphism.→ The injectivity is clear. The surjectivity follows from a standard→ result in distribution theory, namely, that a distribution with point support is a differential operator (see [52, Prop. 1, p. 242]). To prove the multiplicativity, we need to show that (XeYe)e = Xee ∗ Yee (4.1.6) where ∗ denotes the convolution product. The paring of Xee ∗ Yee with f in C (A) is deﬁned as

∞ hXee ∗ Yee, fi := hXee ⊗ Yee, fî, where fîs a smooth function on A × A such that ˆ ∗ f(x, y) = f(xy) = `xf(y), and the ith factor (i = 1, 2) of Xee ⊗ Yee pairs with fˆby viewing it as a function that depends only on the ith variable; thus, hXee ⊗ Yee, fî = hXee,FY,fi where FY,f is a smooth function on A whose value at x ∈ A is ∗ d FY,f(x) := hYee, `xfi = f(x exp(tY)). (4.1.7) dt 0 Thus,

d hXee ⊗ Yee, fˆi = hXee,FY,fi = FY,f(exp(tX)) dt 0

d d = f(exp(tX) exp(sY)) = h(XeYe)e, fi. dt 0 ds 0 This proves Equation 4.1.6, which implies that the linear isomorphism 4.1.5 is multiplicative. Composing the isomorphisms 4.1.4 and 4.1.5, we get an algebra isomorphism 0 δ := ε ◦ τ : U(a) Ee(A). (4.1.8) This gives us a second identiﬁcation for U(a), namely, as the algebra of distributions on A with point support→ on the identity. Applying this to the case A = G, we see that U(a) = U(g) can be viewed as

52 the algebra of distributions on G with support {e}. Applying to the case A = g, we see that U(a) = S(g) can be viewed as the algebra of distributions on g with support {0}.

4.1.9 POINCARÉ-BIRKHOFF-WITT ISOMORPHISM. Let T(g) be the tensor algebra of g. Let I be the ideal in T(g) that is generated by the elements of the form X ⊗ Y − Y ⊗ X. (4.1.10) Then T(g)/I is the symmetric algebra S(g). On the other hand, let J be the ideal in T(g) generated by the elements of the form X ⊗ Y − Y ⊗ X − [X, Y]. (4.1.11) Then T(g)/J is the universal enveloping algebra U(g). It is customary to denote X1 ⊗ · · · ⊗ Xn + I in S(g) or X1 ⊗ · · · ⊗ Xn + J in U(g) simply as X1 ··· Xn. In any case, the elements X1 ··· Xn, where Xi (i ∈ {1, . . . , n}, n ∈ N) are vectors in g, generate S(g) and U(g) as vector spaces. Consider the symmetrization map PBW : S(g) U(g), 1 X ··· X 7 X ··· X . (4.1.12) 1 n n! σ(1) σ(n) → σ∈S Xn Here Sn denotes the symmetric→ group of degree n. It was demonstrated by H. Poincaré [80], G. Birkhoff [14], and E. Witt [100] that the above map is a vector space isomorphism; hence it is called the Poincaré-Birkhoff-Witt isomorphism. Here we review a proof for the Poincaré-Birkhoff-Witt isomorphism using distribution theory. Start by considering the algebra isomorphism 4.1.8 for A = g and A = G, respectively: ∼ 0 δg : S(g) − E0(g), ∼ 0 δG : U(g) − Ee(G). 0 → Here E0(g) is the algebra of distributions on g with support {0}, and 0 → Ee(G) is the algebra of distributions on G with support {e}. Next, consider the exponential map exp : g G, which is a local diffeomorphism near 0. Since functions pull-back along smooth maps of manifolds, their dual objects, the distributions,→ push-forward. More 0 precisely, the pushforward of Λ in E0(g) along the exponential map 0 is the distribution exp∗Λ in Ee(G) deﬁned by ∗ hexp∗Λ, fi := hΛ, exp fi for smooth functions f on G. Because the exponential map is a local diffeomorphism, the push-forward map 0 0 exp∗ : E0(g) Ee(G) (4.1.13) is a vector space isomorphism. Then, demonstrating that the following diagram is commutative yields a→ proof for the Poincaré-Birkhoff-

53 Witt isomorphism:

S(g) PBW / U(g) ∼ δg δG ( )

∼ 4.1.14 0 ∼ 0 E (g) / Ee(G) 0 exp∗ Let us verify this beginning with the simple case. Let X be a vector in g, and view it as a generator in S(g). Take its kth power Xk in S(g). We wish to check that k k exp∗(δg(X )) = δG(PBW(X )). (4.1.15) k k Since δg and δG are multiplicative and PBW(X ) = X , we may just show that k k exp∗(δg(X) ) = δG(X) . (4.1.16) Let f be a smooth function on G. By the deﬁnition of the pushforward map exp∗, k k ∗ hexp∗(δg(X) ), fi = hδg(X) , exp fi. (4.1.17) By the deﬁnition of the convolution product of distributions, we have, for the right-hand side of Equation 4.1.17,

k ∗ ∗ hδg(X) , exp fi = hδg(X) ⊗ · · · ⊗ δg(X), exp\fi, (4.1.18) where exp\∗f is a smooth function gk deﬁned by

∗ ∗ exp\f(X1,...,Xk) = exp f(X1 + ··· + Xk),

∗ and the ith factor of δg(X) ⊗ · · · ⊗ δg(X) pairs with exp\f by viewing it as a function that depends only on the ith variable. Applying Equation 4.1.7 iteratively, we get

∗ hδg(X) ⊗ · · · ⊗ δg(X), exp\fi

d d ∗ = ··· exp f(t1X + ··· + tkX) dt dt 1 t1=0 k tk=0

d d = ··· f(exp(t1X + ··· + tkX)) dt dt 1 t1=0 k tk=0

d d = ··· f(exp(t1X) ··· exp(tkX)) dt dt 1 t1=0 k tk=0

= (Xe ··· Xe)ef = hδG(Xe ··· Xe), fi

= hδG(Xe) ∗ · · · ∗ δG(Xe), fi. (4.1.19) Equations 4.1.17–4.1.19 prove Equation 4.1.16, which is equivalent to Equation 4.1.15. For the general case, just put X = t1X1 + ··· + tnXn into Equation 4.1.15, where t1, . . . , tn are indeterminates, and compare both sides of the resulting equation.

54 4.1.20 Let Cc (M) be the space of compactly supported smooth functions on a manifold∞ M. Suppose the Lie group A acts on M on the left. This induces a G-action on Cc (M) by Equation 4.1.2. Since dis- 0 tributions are dual to functions, we∞ deﬁne the G-action on D (M) in the same manner as in Equation 2.2.7; that is, for g ∈ A and ϕ ∈ D0(M), we deﬁne the distribution g · ϕ on M by setting its paring with f ∈ Cc (M) as −1 ∞ hg · ϕ, fi := hϕ, g · fi. Now consider the G-action on g by the adjoint representation, 0 and on G by conjugation. These induce G-actions on E0(g) and 0 Ee(G) as explained above; we shall denote their G-invariant parts, 0 G 0 G respectively, by E0(g) and Ee(G) . The linear isomorphism exp∗ 0 G 0 G then maps E0(g) onto Ee(G) . (This owes to Equation 2.2.6.) Mean- 0 G g while, the preimage of E0(g) under the map δg is S(g) . Likewise, 0 G the preimage of Ee(G) under δG is Z(g). So the commutative diagram 4.1.14, restricted to the invariant parts, gives us

g PBW / S(g) ∼ Z(g) ∼ δg δG ( )

∼ 4.1.21 0 G ∼ 0 G E (g) / Ee(G) 0 exp∗

4.1.22 DUFLO ISOMORPHISM. The reason we gave an interpretation of the Poincaré-Birkhoff-Will isomorphism in the language of distribution is that the Duflo isomorphism is simpler to state in terms of distributions rather than that of differential operators. In short, the Duflo isomorphism is a modification of the restricted Poincaré- Birkhoff-Witt isomorphism PBW : S(g)g Z(g) so that it becomes an algebra isomorphism. The corresponding modification of exp∗ is (see the diagram 4.1.21) → exp∗ ◦ j (4.1.23) where j denotes the multiplication by the function

sinh adX /2 j(X) = det1/2 . (4.1.24) adX /2 Thus, the Duflo isomorphism is the algebra isomorphism Duf = S(g)g Z(g) defined by −1 → Duf = (δG) ◦ (exp∗ ◦ j) ◦ δg. (4.1.25) When g is a complex semisimple Lie algebra, the Duflo isomor-

55 phism is (the inverse of) the Harish-Chandra isomorphism1. Du- flo [32] generalized the isomorphism S(g)g ' Z(g) for all finite- dimensional Lie algebra and proved it using Kirillov’s orbit method. Alekseev and Meinrenken [2] gave an algebraic proof for the Duflo isomorphism for quadratic Lie algebras. A sketch of their proof and the algebraic description of the Duflo isomorphism is presented in Section 5.3.19.

4.1.26 S(g) ASTHEASSOCIATEDGRADEDALGEBRAOF U(g). Let I and J be the two ideals in T(g) that we defined in Section 4.1.9 so that T(g)/I = S(g) and T(g)/J = U(g). Imposing the relation defined by I on an element in the homogeneous component T k(g) of T(g) of degree k yields again an element in T k(g), whereas the relation defined by J could yield an element in T `(g) with ` < k. Thus, the grading on the tensor algebra T(g) induces a graded algebra structure on the symmetric algebra: M S(g) = Sk(g), k∈Z where Sk(g) = T k(g)/I. But the same grading on T(g) induces a filtered algebra structure on the universal enveloping algebra: [ U(g) = Uk(g), k∈Z Lk ` where Uk(g) = ( `=0 T (g))/J. For any filtered algebra A = S A (over a field), its associated n∈Z n graded algebra is defined as M gr A = An/An−1, n∈Z where the product structure is defined by

Ak/Ak−1 × A`/A`−1 Ak+`/Ak+`−1 (x + Ak−1, y + A`−1) 7 xy + Ak+`−1. The canonical projection → → σk : Ak Ak/Ak−1

is called the symbol map of order k. The maps σk, k ∈ Z, collectively → 1 The Harish-Chandra isomorphism [50, Lem. 36, Lem. 38] for a complex semisimple Lie algebra g is usually presented as an algebra isomorphism γ : Z(g) −∼ S(h)W where h is a Cartan subalgebra of g (which means that h is a maximal abelian W subalgebra of g such that ad(h) is simultaneously diagonalizable) and S(h)→ is the subalgebra of S(h) consisting of elements that are invariant under the action of the Weyl group (the group generated by the reﬂections with respect to the root hyperplanes like the ones we have seen for the Lie algebra of a maximal torus in a compact Lie group). The restriction of a polynomial function on g∗ to h∗ induces a surjective map S(g) S(h) whose restriction to the ad(g)-invariants yields an algebra isomorphism π : S(g)g −∼ S(h)W . So π−1 ◦ γ yields the algebra ∼ isomorphism Z(g) − S(g)g, to→ which the Duﬂo isomorphism is the inverse. For more on the Harish-Chandra isomorphism,→ see [59, § 23.3] or [62, Thm. 5.44]. →

56 deﬁne the symbol map M σ : A gr A = Ak/Ak−1. k∈Z (The above deﬁnitions appear→ in [27, 85].) The point we are trying to make is that the Poincaré-Birkhoff- Witt isomorphism implies that S(g) is isomorphic to gr U(g). To see n this, choose a basis { Xi }i=1 for g. Then the following set of simple tensors,

: k Bk = { Xi1 ⊗ · · · ⊗ Xik ∈ T (g) | 1 6 i1 6 ··· 6 ik 6 n }, yields, under the quotient map T(g) T(g)/I, a basis for the homogeneous component of S(g) of degree k; that is, we have a vector space isomorphism → B Sk(g), SpanR k Xi1 ⊗ · · · ⊗ Xik 7 Xi1 ··· Xik . → Here Span Bk denotes the R-linear span of Bk. Meanwhile, the R → same simple tensors Xi1 ⊗· · ·⊗Xik can be used to deﬁne the following linear map: Span B U (g)/U (g), R k k k−1 (4.1.27) Xi1 ⊗ · · · ⊗ Xik 7 Xi1 ··· Xik + Uk−1(g). This map is clearly surjective.→ To see that it is injective, note that → it factors through the quotient map πJ : T(g) T(g)/J. Now per- muting the vectors that constitute the simple tensor Xi1 ⊗ · · · ⊗ Xik in Bk does not give another simple tensor in B→k. As a consequence, the relations in U(g) deﬁned by the ideal J cannot be applied to a nonzero linear combination of elements in πJ(Bk) to obtain an element of Uk−1(g). Therefore, the map 4.1.27 is injective. Hence, we have a linear isomorphism: M k M S (g) Uk(g)/Uk−1(g), k∈Z k∈Z (4.1.28) X ··· X 7 X ··· X + U (g). i1 ik → i1 ik k−1 To prove that this map is multiplicative, we must show that → (Xi1 ··· Xik )(Xj1 ··· Xj` ) + Uk+`−1(g)

= (Xj1 ··· Xj` )(Xi1 ··· Xik ) + Uk+`−1(g). This would be true if gr U(g) is commutative, which is the case since the commutator of two generators X and Y in g is zero, that is,

XY = YX mod U1(g). To sum up, S(g) and gr U(g) are isomorphic as algebras.. Notice that the isomorphism 4.1.28 is equal to the composition σ ◦ PBW, where σ : U(g) gr U(g) is the symbol map. So we have a

→

57 commutative diagram:

S(g) PBW / U(g) (4.1.28) ∼ u σ gr U(g) Thus, under the identiﬁcation of S(g) as the associated graded algebra of U(g), we may say that the Poincaré-Birkhoff-Witt isomorphism is the inverse of the symbol map.

4.2 THEASYMPTOTICEXPANSIONOF THEHEATKERNEL

4.2.1 The perspective we took in the previous section was that of distribution theory. We now return to the differential operator point of view.

4.2.2 DEFINITION. Let D(g) be the algebra of constant coefﬁcient differential operators on g. Let D(G) be the algebra of left-invariant differential operators on G. Consider the G-action on g by the adjoint representation, and on G by conjugation; this induces G-actions on D(g) and D(G) (see Section 4.1.1); denote the G-invariant parts by D(g)G and D(G)G, respectively. (Then D(G)G is the algebra of bi- invariant differential operators on G as before.) Owing to the algebra isomorphism 4.1.4, we can identify D(g)G with S(g)g and D(G)G ∼ with Z(g). The Duﬂo isomorphism Duf : S(g)g − Z(g) then induces the algebra isomorphism ∼ Duf : D(g)G − D(G)G → so that we have the following commutative diagram: → S(g)g Duf / Z(g)

τg τG G Duf G δg D(g) / D(G) δG (4.2.3)

εg εG exp ◦j Ó 0 G ∗ / 0 G E0(g) Ee(G)

Here the maps τg and τG are provided by (the appropriate restrictions of) the isomorphism 4.1.4 for A = g and A = G, respectively. Likewise, the maps εg and εG are provided by the isomorphism 4.1.5. The arrows in the above diagram are all algebra isomorphisms.

4.2.4 PROPOSITION. Let ∆g and ∆G be the Laplacians on g and G, respectively. Then 2 Duf(∆g) = ∆G − kρk . (4.2.5) Here ρ is half the sum of the positive roots of G, and k · k is the norm induced by the bi-invariant metric on G.

58 Proof. By the deﬁnition of the map Duf, −1 Duf(∆g) = τG ◦ Duf ◦ τg (∆g). (4.2.6)

Let {X1,...,Xn} be any orthonormal basis for g. The map τg maps Xi to the partial derivative ∂i on g with respect to the vector Xi. So n n 2 τg( i=1 XiXi) = i=1 ∂i , which is ∆g. Therefore, n P P −1 τg (∆g) = XiXi. (4.2.7) i=1 X It is known that the image of this element under the Duﬂo isomorphism satisﬁes n 1 Duf X X = Ω + tr (Ω), (4.2.8) i i 24 g i=1 X where Ω is the Casimir element in Z(g) and trg(Ω) is the trace of the Casimir for its action on g arising from the adjoint representation ad : g End(g). An algebraic proof of Equation 4.2.8 can be found in the work of A. Alekseev and E. Meinrenken [3]. We shall review their theory→ in Chapter 5. In particular it is Equation 5.4.23 with g = k that gives Equation 4.2.8. Owing to a result of B. Kostant [65, Eq. 1.85], 1 tr Ω = −kρk2. (4.2.9) 24 g Inserting this into Equation 4.2.8, n 2 Duf XiXi = Ω −kρk . (4.2.10) i=1 X So, by Equations 4.2.6, 4.2.7 and 4.2.10, we have 2 2 Duf(∆g) = τG(Ω −kρk ) = τG(Ω) − kρk .

Since τG(Ω) = ∆G (Equation 2.2.18), we have 2 Duf(∆g) = ∆G − kρk .

4.2.11 DEFINITION. Recall that the exponential map is a local diffeomorphism near 0 in g, that is, there is an open neighborhood V of 0 that is mapped diffeomorphically onto exp(V) =: U under the exponential map. Denote the diffeomorphism of V onto U via the exponential map by expV : V U. (4.2.12) For an invariant differential operator D on G, let Dexp be the constant coefficient differential operator→ on V such that, exp ∗ −1 ∗ D φ = expV (D(expV ) φ). Since Dexp is a constant coefficient differential operator, it makes sense to extend it to a constant coefficient differential operator on g.

59 So we have a linear map D(G) D(g), (4.2.13) D 7 Dexp. → Remark. (1) Suppose f is a smooth function on G. Let f be its → U restriction to U, and set exp ∗ f := expV fU. (4.2.14) Then, Dexpfexp = (Df)exp. (4.2.15) In a word, fexp and Dexp are the expressions for f and D under a local exponential chart near the identity in G.

(2) Owing to the algebra isomorphism 4.1.5, we can identify D(g) 0 0 with E0(g) and D(G) with Ee(G). Then the inverse of the vector space isomorphism 4.1.13 induces a linear isomorphism −1 ∼ expg∗ : D(G) − D(g) (4.2.16) so that we have the following commutative diagram: → exp−1 D(g) o g∗ D(G) ∼ εg εG (4.2.17) ∼ E0 (g) o E0 (G) 0 −1 e exp∗

−1 The linear isomorphism expg∗ is precisely the map 4.2.13; −1 exp expg∗ (D) = D . (4.2.18)

(3) Consider the G-action on g by the adjoint representation, and on G by conjugation. Then the commutative diagram 4.2.17, restricted to the invariant parts, yields

exp−1 D(g)G o g∗ D(G)G ∼ εg εG (4.2.19) ∼ E0 (g)G o E0 (G)G 0 −1 e exp∗

1/2 sinh ad /2 4.2.20 PROPOSITION. Let j(X) = det ( X ). We have adX /2 exp −1 Duf(∆g) = j ◦ ∆g ◦ j.

Remark. The above equation is an equality of differential operators; j and j−1 are to be read as the multiplication operators by the respective functions.

60 Proof. By Equation 4.2.18, exp −1 Duf(∆g) = expg∗ Duf(∆g) . Then, by the commutative diagram 4.2.19, exp −1 −1 Duf(∆g) = εg ◦ exp∗ ◦ εG Duf(∆g) . (4.2.21) Now, by the commutative diagram 4.2.3, εG Duf(∆g) = exp∗ ◦ j ◦ εg(∆g) = exp∗ ◦ j(∆g,0) = exp∗ (∆g ◦ j)0 . (4.2.22) Inserting this into Equation 4.2.21, we get exp −1 Duf(∆g) = εg (∆g ◦ j)0 . This implies that exp Duf(∆g) = φ ◦ ∆g ◦ j (4.2.23) for some smooth function φ on g such that φ(0) = 1. It remains to show that φ = 1/j. Since Duf is an algebra isomorphism, we have 2 2 Duf(∆g) = (Duf ∆g) . Hence, 2 exp exp exp Duf(∆g) = Duf(∆g) ◦ Duf(∆g) . (4.2.24) Repeating the argument we gave for Equation 4.2.23 word for word, 2 except replacing ∆g with ∆g, we get 2 exp 2 Duf(∆g) = φ ◦ ∆g ◦ j. (4.2.25) Meanwhile, by Equation 4.2.23, exp exp Duf(∆g) ◦ Duf(∆g) = φ ◦ ∆g ◦ (jφ) ◦ ∆g ◦ j. (4.2.26) So, by Equations 4.2.24–4.2.26, 2 ∆g = ∆g ◦ ψ ◦ ∆g (4.2.27) where ψ := jφ. We need to show that ψ ≡ 1. Since ψ(0) = 1, it is sufﬁcient to show that grad ψ ≡ 0, where grad is the usual gradient operator on the smooth functions on the Euclidean space g. For an arbitrary vector Y in g, deﬁne the smooth functions sY and cY on g by sY(X) = sinhY, Xi and cY(X) = coshY, Xi. Then, 2 ∆g(sY) = −kYk sY.

2 4 ∆g(sY) = kYk sY. (4.2.28)

61 Whereas,

∆g ◦ ψ ◦ ∆g(sY) 2 = −kYk ∆g(ψsY) 2 2 = −kYk ∆g(ψ)sY + 2hgrad ψ, grad sYi + ψ∆g(sY) 2 2 = −kYk ∆g(ψ) − kYk ψ sY + 2hgrad ψ, YicY. (4.2.29) Hence, by Equations 4.2.27–4.2.29, we have 2 2 2 kYk ∆g(ψ) − kYk ψ + kYk sY = 2hgrad ψ, YicY.

Since sY and cY are linearly independent, we have hgrad ψ, Yi = 0. Because Y is arbitrary, we have grad ψ ≡ 0 as desired.

Combining Propositions 4.2.4 and 4.2.20 gives us the following known result (see [55, Thm. 3.15, p. 273]):

4.2.30 COROLLARY. The Laplacian ∆G on G, under the local exponential chart near the identity, takes the following form: exp −1 2 ∆G = j ◦ ∆g ◦ j + kρk .

4.2.31 PROPOSITION. Let pt be the heat convolution kernel of Duf(∆g). (For the deﬁnition of the heat convolution kernel, see Section 2.1.9.) exp ∗ Then pt := exp pt has the asymptotic expansion 1 pexp ∼ h t t j for t 0+, valid in some neighborhood of 0 ∈ g, where ht is the −kXk/4t − dim(g)/2 Gaussian kernel on g, that is, ht(X) = e (4πt) . In other words,→ the coefﬁcients of the asymptotic expansion 2.1.15, for this case, are a0(x) = 1/j(log(x)) and an(x) = 0 for n ∈ N.

Remark. The reason ht/j is not the exact heat kernel is that the exponential map fails to be a global diffeomorphism.

Proof. Writing the heat equation, (∂t + Duf(∆g))pt = 0, in the exponential chart near the identity in G, we get exp exp (∂t + Duf(∆g) )pt = 0. exp : i Let st = ht i=0 ait be the asymptotic expansion for pt . It is the formal solution to P∞ exp (∂t + Duf(∆g) )st = 0. (4.2.32) By Proposition 4.2.20, the differential equation 4.2.32 is equivalent to 1 (∂ − ∆ )js = 0. j t g t

We need to show that st = ht/j satisﬁes this differential equation. But that is easily seen from the fact that ht satisﬁes the heat equation (∂t − ∆g)ht = 0.

62 4.2.33 LEMMA. Let G be a compact connected Lie group equipped with a bi-invariant metric h , i. The value of the scalar curvature S of G is given by 1 S = − tr (Ω) = 6kρk2, (4.2.34) 4 g where ρ is half the sum of the positive roots of G.

Proof. As we pointed out in Section 2.2.14, the Riemannian connection ∇ on G satisfies ∇ Y = 1 [X, Y]. So the curvature tensor R Xe e 2 e e satisfies R(X, Y)Z = ∇ ∇ Z − ∇ ∇ Z − ∇ Z (by definition) e e e Xe Ye e Ye Xe e [X,e Ye] e 1 1 1 = [X,e [Y,e Ze]] − [Y,e [X,e Ze]] − [[X,e Ye], Ze]. 4 4 2 Applying the Jacobi identity to the above, we get 1 R(X,e Ye)Ze = − [[X,e Ye], Ze]. 4 Then, the Riemann curvature tensor Rm satisfies Rm(X,e Y,e Z,e Wf) = hR(X,e Ye)Z,e Wfi (by definition) 1 = − h[[X, Ye], Ze], Wfi 4 1 = − h[[X, Y],Z],Wi, (4.2.35) 4 where the last equality owes to the left-invariance of the metric. Now the scalar curvature S is, by definition, the trace of the Ricci tensor; dim g this means, in terms of an orthonormal basis { Xi }i=1 for g, dim g S = Rm(Xei, Xej, Xej, Xei). (4.2.36) i,j=1 X So, by Equation 4.2.35,

dim g 1 S = − h[[X ,X ],X ],X i 4 i j j i i,j=1 X dim g 1 = − h[ad ◦ ad (X ),X i 4 Xj Xj i i i,j=1 X dim g 1 1 = − had(Ω)X ,X i = − tr (Ω). 4 i i 4 g i,j=1 X The proves the ﬁrst equality in Equation 4.2.34. The second equality follows from Equation 4.2.9.

4.2.37 THEOREM. Let G be a compact connected Lie group equipped with a bi-invariant metric. Let kt be the heat convolution kernel of the

63 exp ∗ Laplacian on G. Then kt := exp kt has the asymptotic expansion

tS/6 n exp e 1 1 S n kt ∼ ht = ht t (4.2.38) j ∞ j n! 6 n=0 X for t 0+, valid in a neighborhood of 0 ∈ g. Here j is the function deﬁned by the power series 4.1.24, S is the scalar curvature of G, and ht −kXk2/4t − dim(g)/2 is the→ Gaussian kernel on g, that is, ht(X) = e (4πt) . Proof. By Proposition 4.2.4, we have 2 ∆G = Duf(∆g) + kρk . This implies that 2 et∆G = etkρk et Duf(∆g). Then, by Lemma 4.2.33,

et∆G = etS/6et Duf(∆g). This implies that tS/6 kt = e pt, where pt is the heat convolution kernel of Duf(∆g). So the asymptotic expansion 4.2.38 follows from Proposition 4.2.31.

4.2.39 COROLLARY. Let G be a compact connected Lie group equipped with a bi-invariant metric. Let S be the scalar curvature of G. The heat trace Z(t) = tr(et∆G ) has the asymptotic expansion vol(G) Z(t) ∼ etS/6 (4.2.40) (4πt)dim G/2 for t 0+.

Proof. This follows from Theorem 4.2.37 and Equation 2.1.14. → Example. Let G = SU(2). Take the bi-invariant metric on G generated by the negative of the Killing form on the Lie algebra of G. We examined this case in Section 3.2, and calculated that (Equations 3.2.10 and 3.2.23) 1 √ kρk2 = , vol(G) = 32 2π2. 8 So, by Lemma 4.2.33 and Corollary 4.2.39, the heat trace Z(t) has the following asymptotic expansion: √ 4 2π Z(t) ∼ et/8. t3/2 Remark. H. P. McKean and I. M. Singer [74] calculated the asymptotic expansion for the heat trace, using differential geometric methods, up to O(t3−n/2) for any closed Riemannian manifold, where n is the dimension of the manifold. Our result (Corollary 4.2.39) is consistent with theirs. H. D. Fegan, in [39], obtained a formula for the

64 exact heat kernel on a simply connected, compact, semisimple Lie group; but S. Zelditch [101] claims that it is erroneous. H. D. Fegan, using his heat kernel formula, also derived the asymptotic expansion for the heat trace; somehow, it is consistent with our result.

65 THEQUANTUMWEILALGEBRA5

O far our focus has been on the Laplacian. We now shift our S attention to an operator that is in a sense a “square root” of the Laplacian, namely, the Dirac operator. As P. A. M. Dirac [29] no- ticed, the vector bundle on which the Dirac operator acts cannot, in general, be the trivial line bundle, and we must consider the spinor bundles. A natural algebraic model for the space of equivariant differential operators acting on the sections of the spinor bundles over homogeneous spaces is the quantum Weil algebra introduced by A. Alekseev and E. Meinrenken [2,3]. This chapter is a brief account on their theory. As usual, G denotes a compact connected Lie group with Lie algebra g, equipped with an invariant inner product on g (that is, the ad(g)-action is antisymmetric).

Remark. Following the custom, we shall call a differential operator on the space of sections of a vector bundle F as a differential operator on the vector bundle F.

5.1 CLIFFORDALGEBRAS

5.1.1 A prerequisite for understanding the quantum Weil algebra is the theory of Clifford algebras. Let us go over some relevant facts and the conventions we shall use.

5.1.2 CLIFFORD ALGEBRA OF AN INNER PRODUCT SPACE. Let V be a vector space over R equipped with an inner product h , i. Let T(V) be the tensor algebra of V. Let J(V) denote the ideal in T generated by the elements of the form vw + wv − hv, wi. (5.1.3)

66 The Clifford algebra generated by (V, h , i) is

Cl(V) := T(V)/J(V). There is a universal property that the Clifford algebra inherits from the tensor algebra; namely, if there is a linear map ϕ from V to a unital (associative) algebra A over R that satisﬁes ϕ(v)ϕ(v) = hv, vi/2, then there is a unique algebra homomorphismϕ ˜ : Cl(V) A such that ϕ = ϕ˜ ◦iV where iV is the inclusion map of V into Cl(V). By the usual argument for universal objects, Cl(V) is isomorphic→ to n the Clifford algebra Cl(n) generated by the Euclidean space R .

Remark. Note that the vectors v and w in V, as generators of Cl(V), satisfy the relation vw + wv = hv, wi. (5.1.4) The traditional convention is to use the relation vw + wv = hv, wi with equal to 2 or −2. The speciﬁc choice for the value of bears no weight for our purposes. We do point out, though, that because we chose = 1, the top order part (or the principal symbol) of the square of the Dirac operator will be equal to that of the Laplacian times 1/2.

5.1.5 SPINORS. We deﬁne the complexiﬁcation of Cl(V) as Cl(V) := Cl(V) ⊗ C. Since Cl(V) ' Cl(n) for some nonnegative integer n, we have

Cl(V) ' Cl(n) ⊗ C =: Cl(n). According to the general theory of Clifford algebras (see [43, p. 13]),

2k EndC(C ), if n = 2k; Cl(n) ' 2k 2k (5.1.6) EndC(C ) ⊕ EndC(C ), if n = 2k + 1. These isomorphisms give rise to an irreducible representation of k Cl(n) on the 2 -dimensional complex vector space with k = bn/2c. This representation space is called the space of spinors for Cl(n) (or n-spinors, for short); we shall denote it by Sn. It is clear from the isomorphism 5.1.6 that, when n = 2k, there is only one way Cl(2k) can act on S2k. As a consequence, any Cl(2k)-module E is of the form E = S2k ⊗ W (5.1.7) where W is some auxiliary complex vector space on which Cl(2k) acts trivially; put in another way,

W ' HomCl(2k)(S2k,E).

67 When n = 2k + 1, it turns out that there are two ways Cl(2k + 1) can act on S2k+1; see [43, p. 13] for details.

n 5.1.8 Let { ei }i=1, n = dim(V), be an orthonormal basis for V. Then

the simple tensors of the form ei1 ⊗ · · · ⊗ eik , k ∈ N, together with 1, form a basis for the tensor algebra T(V). It is customary to write the

image of ei1 ⊗· · ·⊗eik under the quotient map T(V) T(V)/J(V) =

Cl(V) as ei1 ··· eik . Among these images, the ones that are not zero are the following: For each I ⊆ {1, . . . , n}, →

1, for I = ∅; e := (5.1.9) I e ··· e , for I = {i , . . . , i } with i < ··· < i . i1 ik 1 k 1 k

The collection { eI | I ⊆ {1, . . . , n}} is a linear basis for Cl(V). 2n As a consequence, we have an isomorphism Cl(V) ' R as vector spaces. This induces a differentiable structure on Cl(V). Ac- cording to a general fact regarding the group of units in a ﬁnite- dimensional algebra (see, for instance, [28, § 16.9.3]) the group Cl(V)× of units in Cl(V) is an open subset of Cl(V) and is a Lie group under the inherited submanifold structure.

5.1.10 Cl(V) AS A FILTERED ALGEBRA. The natural grading on T(V) induces a filtration on Cl(V), that is, k M q Clk(V) := T (V) J(V), k ∈ Z, q=− q gives a filtration for Cl(V). (In∞ the above definition, we set T (V) = {0} if q is negative.) The basis elements eI defined by Equation 5.1.9 have filtration order |I|. Thus, { e : |I| k } = (V). SpanR I 6 Clk (5.1.11) Moreover, we have a canonical linear isomorphism σ : Span { e : |I| = k } Cl (V)/Cl (V), k R I k k−1 (5.1.12) eI 7 eI + Clk−1(V), → for each k ∈ Z; collectively, we have a vector space isomorphism M→ σ : (V) (V)/ (V). Cl Clk Clk−1 (5.1.13) k∈Z This is the symbol map (see→ page 57) from Cl(V) to its associated graded algebra gr Cl(V).

5.1.14 Cl(V) ASASUPER-ALGEBRA. The tensor algebra T(V) becomes a super-algebra1 by reducing its natural Z-grading modulo 2Z. Since the relation 5.1.4 respects this Z/2Z-grading, the Clifford algebra

1 ‘Super’ is synonymous to ‘Z/2Z-graded’.

68 Cl(V) is also a super-algebra: Cl(V) = Cl0(V) ⊕ Cl1(V).

In terms of the basis elements eI (Equation 5.1.9), we have 0(V) = { e : |I| ≡ 0 2 } , Cl SpanR I mod 1(V) = { e : |I| ≡ 1 2 } . Cl SpanR I mod

We deﬁne the super-commutator [ , ]s on Cl(V) as, for homogeneous elements x and y, deg(x) deg(y) [x, y]s = xy − (−1) yx.

5.1.15 CHEVALLEY IDENTIFICATION. Let ∈ [0, 1]. Deﬁne Cl(V, ) as the Clifford algebra generated by (V, h , i). If = 1, we get back Cl(V). If = 0, then we have the exterior algebra ∧(V). n As before, let { ei }i=1 be an orthonormal basis for V with respect to the original inner product h , i. Deﬁne, for each I ⊆ {1, . . . , n}, the elements eI in Cl(V, ) as

1, for I = ∅; e := (5.1.16) I e ··· e , for I = {i , . . . , i } with i < ··· < i . i1 ik 1 k 1 k 1 These elements form a linear basis for Cl(V, ). Note that eI is what we deﬁned earlier as eI (Equation 5.1.9). In any case, we have a vector space isomorphism: Cl(V, ) Cl(V), eI 7 eI. When = 0, this vector space isomorphism→ is → q : ∧(V) Cl(V), (5.1.17) ei1 ∧ ··· ∧ eik 7 ei1 ··· eik . That q is a vector space isomorphism→ was ﬁrst stated and proved by C. Chevalley [25, Thm. II.1.6, p. 41].→ Following A. Alekseev and E. Meinrenken [3], we call this map the Chevalley (quantization) map. Note that, since

eiej = −ejei + δij, (5.1.18) where δij is the Kronecker delta symbol, we have 1 e ··· e = sgn(σ) e ··· e . i1 ik k! σ(1) σ(k) σ∈S Xk The right-hand side is the image of the alternating tensor 1 sgn(σ) e ⊗ · · · ⊗ e (5.1.19) k! σ(1) σ(k) σ∈S Xk under the canonical projection πCl : T(V) T(V)/J(V) = Cl(V). Now the tensors of the form 5.1.19 span the subspace Alt(V) of the →

69 alternating tensors in T(V). It is a standard fact in algebra (see [31, Prop. 40, p. 453]) that Alt(V) is linearly isomorphic to ∧(V) by the antisymmetrization map: 1 Alt(v ∧ ··· ∧ v ) = sgn(σ) v ⊗ · · · ⊗ v . (5.1.20) 1 k k! σ(1) σ(k) σ∈S Xk Here vi’s are arbitrary vectors in V. So we have the following commutative diagram of linear isomorphisms: Alt(V) ⊆ T(V) 8 Alt πCl / ∧(V) q Cl(V) In this sense, the Chevalley map is the antisymmetrization map. It is reasonable to guess that ∧(g) is isomorphic to gr Cl(V) as algebras. Let us check this. The very deﬁnition of the Chevalley map 5.1.12 shows that the kth homogeneous component ∧k(V) of { e : |I| = k } the exterior algebra is mapped bijectively onto SpanR I in the Clifford algebra. So the composition of the Chevalley map with the linear isomorphism 5.1.12 gives us the linear isomorphism σ ◦q ∧k(V) −−−k Cl (V)/Cl (V), k k−1 (5.1.21) eI 7− eI + Clk−1(V). → Collectively, we have the linear isomorphism → σ◦q ∧(V) −− gr Cl(V). (5.1.22) This isomorphism is, in fact, multiplicative (so it is an algebra isomorphism). It is sufﬁcient to check→ for the product of two basis elements ei1 ∧···∧eik and ej1 ∧···∧ej` in ∧(V), where 1 6 i1 < ··· < ik 6 n and 1 6 j1 < ··· < j` 6 n. The image of these two basis elements under the map 5.1.22 are ei1 ··· eik + Clk−1(V) and ej1 ··· ej` + Cl`−1(V), respectively, and their product is ei1 ··· eik ej1 ··· ej` + Clk+`−1(V). So we need to check whether q ei1 ∧···∧eik ∧ej1 ∧···∧ej` = ei1 ··· eik ej1 ··· ej` mod Clk+`−1(V). (5.1.23)

If {i1, . . . , ik} ∩ {j1, . . . , j`} = ∅, then ei1 ∧ ··· ∧ eik ∧ ej1 ∧ ··· ∧ ej` is one of the standard basis element (up to sign) for ∧(V), so q maps it to ei1 ··· eik ej1 ··· ej` . If {i1, . . . , ik} ∩ {j1, . . . , j`} 6= ∅, then ei1 ∧

··· ∧ eik ∧ ej1 ∧ ··· ∧ ej` = 0, but ei1 ··· eik ej1 ··· ej` is in Cl|I|+|J|−2(V) (owing to the relation eiej = −ejei + δij), so Equation 5.1.23 holds. In fact, our calculations show that we have a stronger relation: q ei1 ∧···∧eik ∧ej1 ∧···∧ej` = ei1 ··· eik ej1 ··· ej` mod Clk+`−2(V). (5.1.24) To sum up, we may identify ∧(V) with gr Cl(V). Then the Cheval- ley map is the inverse of the symbol map. Put in another way, we

70 have the following commutative diagram of vector space isomorphisms: q ∧(V) Cl(V) s σ◦q ∼ σ + gr Cl(V)

5.1.25 CLIFFORD MULTIPLICATION ON ∧(V). Since Cl(V) and ∧(V) are linearly isomorphic, we can transfer the product structure of Cl(V) to ∧(V); we deﬁne the Clifford multiplication of x and y in ∧(V) as −1 x · y := q (q(x)q(y)). The very fact that that ∧(V) is the associated graded algebra of Cl(V) implies that the Clifford multiplication is equal to the exterior multiplication modulo terms of lower degree. More precisely, we have, by Equation 5.1.24, k+`−2 M x · y = x ∧ y mod ∧q(V) (5.1.26) q=0 for homogeneous elements x and y of degree k and `, respectively.

5.1.27 DEFINITION. We deﬁne the Clifford commutator [ , ]s on ∧(V) to be the super-commutator with respect to the Clifford multiplication. That means k` [x, y]s = x · y − (−1) y · x for homogeneous elements x and y of degree k and `, respectively.

Remark. By Equation 5.1.26, we have k+`−2 k` M q [x, y]s = x ∧ y − (−1) y ∧ x mod ∧ (V). q=0 The exterior algebra is super-commutative in the sense that x ∧ y − (−1)k`y ∧ x = 0; so k+`−2 M q [x, y]s = 0 mod ∧ (V). (5.1.28) q=0

5.1.29 LIEALGEBRAISOMORPHISM ∧2 (V) ' so(V). Let x be a homogeneous element of ∧(V) of degree 2. If v is a vector in V = ∧1(V) 1 then, by Equation 5.1.28, [x, y]s = 0 modulo ∧ (V); so the Clifford 2 commutator [x, v]s is again a vector in V. Hence, each x in ∧ (V) deﬁnes a linear endomorphism on V by

ads(x): V V, v 7 [x, v]s. → As a simple example, consider x := ei ∧ ej with 1 6 i < j 6 n. →

71 Then −1 −1 [x, v]s = q (q(eiej)q(v)) − q (q(v)q(eiej)) −1 = q (q(eiej)q(v) − q(v)q(eiej)) −1 = q (eiejv − veiej). (5.1.30)

From the relation ejv = −vej + hej, vi, we get

eiejv = −eivej + hej, viei = veiej − hei, viej + hej, viei. Inserting this into Equation 5.1.30, −1 [ei ∧ ej, v]s = q (hej, viei − hei, viej)

= hej, viei − hei, viej. (5.1.31)

Hence, the Clifford commutation with ei ∧ej is a antisymmetric operator on V. By the bi-linearity of the commutator, we concluded that this is true for Clifford commutation with any element in ∧2(V). Therefore, we have an injective linear map 2 ads : ∧ (V) so(V). (5.1.32) We may asks whether this is surjective. The answer is afﬁrmative; in fact, as B. Kostant shows in [64, Thm.→ 8, p. 286], this is a Lie algebra isomorphism as we set the Clifford commutator as the Lie bracket for ∧2(V). A formula for the inverse of the map 5.1.32 is as follows: For T in so(V), 1 n ad−1(T) = T(v ) ∧ vi, (5.1.33) s 2 i i=1 X n i n where { vi }i=1 and { v }i=1 are two basis for V that are dual with j respect to the inner product so that hvi, v i = δij. This formula does not depend on the choice for the basis.

5.1.34 DEFINITION. A super-derivation on a super-algebra A is a linear map δ : A A such that δ(xy) = δ(x)y + (−1)deg(x)nxδ(y) → for homogeneous elements x and y in A, where n is any integer. The derivation is said to be even or odd, according to the parity of n. The following lemma is from [64, Lem. 5, p. 284]:

5.1.35 LEMMA. Let ιv denote the interior product on ∧(V) with respect to v, that is, the odd derivation on ∧(V) generated by

ιvu = hu, vi (5.1.36) for u ∈ ∧1(V). Then, for x in ∧2(V),

[x, v]s = −ιvx. (5.1.37)

With slight abuse of notation, let ιv also denote the odd derivation on

72 Cl(V) generated by Equation 5.1.36. Then,

[qx, v]s = −ιv(qx). (5.1.38)

dim V Proof. Let { ei }i=1 be an orthonormal basis for V. Owing to linearity, it is sufﬁcient to check Equation 5.1.37 for the case x = ei ∧ ej; but that is done by Equation 5.1.31. According to the deﬁnition of the Clifford commutator, Equa- tion 5.1.37 means −1 q [qx, qv]s = −ιvx where [ , ]s now represents the super-commutator on Cl(V). So we have [qx, qv]s = −q(ιvx). Since qu = u for any u in V, the above gives us

[qx, v]s = −ιvx. Hence, to prove Equation 5.1.38, it is left to show that

ιvx = ιv(qx). (5.1.39)

Again, it is sufﬁcient to check for the case x = ei ∧ ej. For ιv(qx), we have ιv(q(ei ∧ ej)) = ιv(eiej).

Then, by the odd derivation property of ιv,

ιv(q(ei∧ej)) = ιv(ei)ej−eiιv(ej) = hei, viej−hej, viei = ιv(ei∧ej).

5.1.40 SPIN GROUP. We deﬁne the spin group Spin(V) and its Lie algebra spin(V) as follows. (This is due to B. Kostant; see § 2, especially Thm. 8, in [64].) Consider the inverse of the map 5.1.32, −1 ∼ 2 ads : so(V) − ∧ (V). (5.1.41) As we have noted in Section 5.1.29, this is a Lie algebra isomorphism where the Lie bracket on ∧2(V) is given→ by the Clifford commutator. The Clifford commutator on ∧(V) was deﬁned so that the Cheval- ley map q intertwines it with the super-commutator on Cl(V); thus, −1 composing ads with q yields a Lie-algebra isomorphism of so(V) onto its image in Cl(V); this image is the Lie algebra spin(V). ∼ so(V) −−−− spin(V) ⊆ Cl(V). −1 (5.1.42) q◦ads dim V Let { ei }i=1 be an orthonormal→ basis for V. Since the set

{ ei ∧ ej | 1 6 i < j 6 dim V } spans ∧2(V), its image under the quantization map

{ eiej | 1 6 i < j 6 dim V } spans spin(V).

73 Now let expCl denote the exponentiation in Cl(V); that is, for x in Cl(V), xn expCl(x) = . ∞ n! n=0 X Then, Spin(V) := expCl(spin(V)).

The Clifford exponential map expCl on spin(V) agrees with the Lie- theoretic exponential map exp : spin(V) Spin(V), owing to the uniqueness of 1-parameter subgroups. As an example, let us calculate expCl(te→iej) where t is a parameter in R. Note that 1 (e e )2 = e e e e = −e e e e = − . i j i j i j i i j j 22 Hence, (−1)k t 2k (−1)k t 2k+1 expCl(teiej) = + 2eiej ∞ (2k)! 2 ∞ (2k + 1)! 2 k=0 k=0 X X = cos(t/2) + 2 sin(t/2)eiej. (5.1.43) As our discussion in Section 5.1.29 shows, for each x in spin(V), the inner derivation it deﬁnes on Cl(V) (with respect to the super- commutator) restricts to a skew-symmetric operator on V. Let us denote this representation by adc : spin(V) so(V). Now deﬁne the Lie group representation Adc : Spin(V) Aut(V) by −1 → Adc(g)(v) = gvg , → where gvg−1 is the product of g, v, and g−1 as elements of Cl(V). Then, adc is the differential of Ads, and we have the following commutative diagram:

Ad Spin(V) c / SO(V) O O

expCl exp spin(V) / so(V) adc

Because adc is a Lie algebra isomorphism, Adc is a covering map (see [92, Thm. 3.25, p. 100]). This covering is evidently not an isomorphism since −1 in Spin(V) acts trivially on v. In fact, it is a double covering; see [43, Prop. (c), p. 16]. Since SO(V) is compact (see [34, § 1.2.A]) and Adc is a covering map with ﬁnite number of sheets, Spin(V) is a compact Lie group.

5.1.44 Because Spin(V) is a subgroup of Cl(V)×, which is contained in Cl(V), the spinor space S for Cl(V) is a complex representation space for Spin(V). This representation is known as the spinor representation of Spin(V). It is a classical result that S, as a Spin(V)-

74 vector space, may be reducible or not, depending on the parity of the dimension of V; if the dimension is odd, then S is irreducible; if the dimension is even, then S is completely reducible to + − S = S ⊕ S , (5.1.45) ± where the irreducible subspaces S are the eigenspaces with eigen- −k values ±(2i) , respectively, for the action of the element e1e2 ··· en where n = dim(V) (see [43, p. 22] for a proof). Since Spin(V) is compact, the spinor space S admits a Spin(V)- invariant inner product. In fact, we can expect more; we may assume that the Spin(V)-invariant inner product h , i is skew-invariant with respect to the action of V in Cl(V), that is, for v ∈ V and (ξ1, ξ2) ∈ S × S, hv · ξ1, ξ2i = −hξ1, v · ξ2i. Moreover, when V is even-dimensional, the decomposition 5.1.45 is an orthogonal decomposition with respect to this inner product. See [43, p. 24] for details.

5.1.46 DEFINITION. Let g be a ﬁnite-dimensional Lie algebra with an inner product h , i that is skew-invariant under the ad(g)-action. Then the adjoint representation yields a Lie algebra homomorphism

g −ad so(g). We shall denote the composition of the above map with the Lie algebra isomorphisms 5.1.41 and 5.1→.42, respectively, by λ : g ∧2(g), γ : g spin(g). In other words, for X in g, → → −1 λ(X) = ads (adX), (5.1.47) γ(X) = q ◦ λ(X). (5.1.48)

Remark. The maps λ and γ are clearly Lie algebra homomorphisms. That means

λ([X, Y]) = [λ(X), λ(Y)]s, (5.1.49)

γ([X, Y]) = [γ(X), γ(Y)]s. (5.1.50)

5.1.51 LEMMA. Let g be a ﬁnite-dimensional Lie algebra over R with an inner product h , i that is skew-invariant under the ad(g)-action. dim g i dim g Let { Xi }i=1 and { X }i=1 be two basis for g that are dual with respect j to the inner product so that hXi,X i = δij. In the following, X and Y are vectors in g, [X, Y]g denotes their Lie bracket in g. (a) The Lie algebra homomorphism λ satisﬁes

adX(Y) = [λ(X),Y]s, (5.1.52)

75 where [ , ]s is the Clifford commutator in ∧(g). And dim g 1 λ(X) = − hX, [X ,X ] iXi ∧ Xj. (5.1.53) 2 i j g i,j=1 X (b) The Lie algebra homomorphism γ satisﬁes

adX(Y) = [γ(X),Y]s, (5.1.54)

where [ , ]s is the super-commutator in Cl(g). And dim g 1 γ(X) = − hX, [X ,X ] iXiXj. (5.1.55) 2 i j g i,j=1 X −1 −1 Proof. By deﬁnition, λ(X) = ads (adX). Now ads was deﬁned so −1 that, for any T ∈ so(g), we would have T(Y) = [ads (T),Y]s. Setting T = adX, we get −1 adX(Y) = [ads (adX),Y]s = [λ(X),Y]s. This proves Equation 5.1.52. Let us deduce Equation 5.1.54 from this. Applying the Chevalley map q to the above equation, we get

q(adX(Y)) = q[λ(X),Y]s.

The left-hand side is just adX(Y). For the right-hand side, recall that the Chevalley map q intertwines the Clifford commutator on ∧(g) with the super-commutator on Cl(g); so

q[λ(X),Y]s = [qλ(X), qY]s = [γ(X),Y]s. Here the ﬁrst commutator is the Clifford commutator in ∧(g), and the other two commutators are the super-commutator in Cl(g). This proves Equation 5.1.54. Next, let us verify Equation 5.1.53. Again, begin with the deﬁni- −1 tion λ(X) = ads (adX). Then, by Equation 5.1.33, dim g dim g 1 1 λ(X) = ad(X)(X ) ∧ Xj = [X, X ] ∧ Xj. (5.1.56) 2 j 2 j g j=1 j=1 X X dim g i Now, for any vector Y in g, we have Y = i=1 hY, XiiX . So dim g P i [X, Xj]g = h[X, Xj]g,XiiX . i=1 X Inserting this into Equation 5.1.56, we get dim g 1 λ(X) = h[X, X ] ,X iXi ∧ Xj. 2 i g j i,j=1 X By the invariance of the inner product, the above can be rewritten

76 as dim g 1 λ(X) = − hX, [X ,X ] iXi ∧ Xj. 2 i j g i,j=1 X This proves Equation 5.1.53. Applying the Chevalley map to both sides of this equation gives us Equation 5.1.55.

5.2 THEQUANTUMWEILALGEBRA

5.2.1 For the rest of this chapter, g is a ﬁnite-dimensional Lie algebra over R equipped with an invariant inner product h , i. (For example, g is the Lie algebra of a compact Lie group endowed with a bi-invariant metric.) Letg ¯ be a copy of g, and for each X in g, denote the corresponding vector ing ¯ by X. The starting point of A. Alekseev and E. Meinrenken’s theory is to consider the super-space (that is, the Z/2Z-graded vector space) g ⊕ g¯, where g is the subspace of odd degree andg ¯ is the subspace of even degree. The quantum Weil algebra W(g) is the unital super-algebra over R generated by the super-space g ⊕ g¯, subject to the following super-commutator relations:

[X, Y]s = hX, Yi, (5.2.2)

[X, Y]s = [X, Y]g, (5.2.3)

[X, Y]s = [X, Y]g. (5.2.4)

Here [ , ]g denotes the Lie bracket in g. Note that g alone generates the Clifford algebra Cl(g) and thatg ¯ alone generates the universal enveloping algebra U(g)¯ . This takes into account Equations 5.2.2 and 5.2.3. Equation 5.2.4 generates a U(g)¯ -action on Cl(g). So, as a vector space, W(g) is isomorphic to Cl(g) ⊗ U(g)¯ . As an algebra, W(g) is a semi-direct product:

W(g) = Cl(g) o U(g)¯ . Each vector X in g generates two inner derivations on W(g):

ιX(·) = [X, ·]s,LX(·) = [X, ·]s. (5.2.5) These are called the contraction and the Lie derivative by X, respectively. The contraction is an odd derivation whereas the Lie derivative is an even one. In terms of the generators, we have

ιXY = hX, Yi, ιXY = [X, Y]g, (5.2.6)

LXY = [X, Y]g,LXY = [X, Y]g. (5.2.7)

77 Augment this collection of derivations with a differential (that is a graded derivation of degree 1) deﬁned by

dX = X, dX = 0. (5.2.8) The contractions, the Lie derivatives, and the differential satisfy the following super-commutator relations [2, Thm. 3.6, p. 145;3, § 3]:

[d, d]s = 0, [ιX, d]s = LX, [LX, d]s = 0,

[LX,LY]s = L[X,Y], [LX, ιY]s = ι[X,Y], [ιX, ιY]s = 0. A. Alekseev and E. Meinrenken [3] refers to this as a g-differential algebra structure (g-da for short). A remarkable fact found by A. Alekseev and E. Meinrenken [3, Prop. 5.1, p. 317] is that the differential is also inner; there is an element D/ in W(g) such that

d(·) = [D/ , ·]s. (5.2.9) dim g In terms of an orthonormal basis { Xi }i=1 for g, dim g 2 D/ = X X − γ(X ) , (5.2.10) i i 3 i i=1 X where γ(Xk) is the image of Xk under the map γ defined in Defi- nition 5.1.46. Because the differential is independent of the choice for the orthonormal basis, Equation 5.2.9 shows that D/ is also independent of the choice for the orthonormal basis. The element D/ is called the cubic “Dirac operator” (cubic with respect to the filtration for W(g) which we shall describe in Section 5.3.1) or the Kostant- Dirac operator in W(g). Another significance of the element D/ lies in the formula for its square; see Equation 5.2.23.

5.2.11 CHANGE OF (EVEN) GENERATORS. Equation 5.1.54 states that

[X, Y]g = [γ(X),Y]s.

According to Equation 5.2.3, the left-hand side is equal to [X, Y]s. So we have [X − γ(X),Y]s = 0. (5.2.12) The element Xb := X − γ(X) is an even element in W(g); it can replace X as an even generator. Denote the new set of even generators by

gˆ := { Xb | X ∈ g } ⊆ W(g).

78 The super-commutator relations satisﬁed by the generators in g and gˆ are

[X, Y]s = hX, Yi, (5.2.13)

[X,b Y]s = 0, (5.2.14)

[X,b Yb]s = [[X, Y]g. (5.2.15) Equation 5.2.13 is from Equation 5.2.2, and Equation 5.2.14 is just Equation 5.2.12. So we only need to verify Equation 5.2.15. Since Xb = X − γ(X) and Yb = Y − γ(Y), we have

[X,b Yb]s = [X, Y]s − [γ(X), Y]s − [X, γ(Y)]s + [γ(X), γ(Y)]s.

Then, by Equations 5.1.50 and 5.2.4,

[X,b Yb]s = [X, Y]g − [γ(X), Y]s − [X, γ(Y)]s + γ([X, Y]g). It is left to check that

− [γ(X), Y]s − [X, γ(Y)]s = −2γ([X, Y]g). (5.2.16) By Equation 5.1.55, dim g 1 [X, γ(Y)] = − hY, [X ,X ] i([X, Xi] Xj + Xi[X, Xj] ). s 2 i j g s s i,j=1 X Applying Equation 5.2.3 to the right-hand side, dim g 1 [X, γ(Y)] = − hY, [X ,X ] i([X, Xi] Xj + Xi[X, Xj] ). s 2 i j g g g i,j=1 X dim g k Since Z = k=1 hZ, XkiX holds for any vector Z in g, we have dim g P 1 [X, γ(Y)] = − hY, [X ,X ] ih[X, Xi] ,X iXkXj s 2 i j g g k i,j,k=1 X dim g 1 − hY, [X ,X ] ih[X, Xj] ,X iXiXk. 2 i j g g k i,j,k=1 X By the invariance of the inner product,

dim g 1 [X, γ(Y)] = − h[Y, X ] ,X ihXi, [X, X ] iXkXj s 2 j g i k g i,j,k=1 X dim g 1 + h[Y, X ] ,X ihXj, [X, X ] iXiXk. 2 i g j k g i,j,k=1 X

79 dim g i Using the fact that hZ, Wi = i=1 hZ, XiihX ,Wi for any vectors Z and W in g, P dim g 1 [X, γ(Y)] = − h[Y, X ] , [X, X ] iXkXj s 2 j g k g j,k=1 X dim g 1 + h[Y, X ] , [X, X ] iXiXk. 2 i g k g i,k=1 X By symmetry, we may exchange X with Y, which gives

dim g 1 [Y, γ(X)] = − h[X, X ] , [Y, X ] iXkXj s 2 j g k g j,k=1 X dim g 1 + h[X, X ] , [Y, X ] iXiXk. 2 i g k g i,k=1 X Therefore,

− [γ(X), Y]s − [X, γ(Y)]s dim g j k = (h[X, Xj]g, [Y, Xk]gi − h[X, Xk]g, [Y, Xj]gi)X X . (5.2.17) j,k=1 X Meanwhile, dim g dim g i j i j −2γ([X, Y]g) = h[X, Y]g, [Xi,Xj]giX X = h[[X, Y]g,Xi]g,XjiX X . i,j i,j X X By the Jacobi identity,

dim g i j −2γ([X, Y]g) = (h[[X, Xi]g,Y]g,Xji + h[X, [Y, Xi]g]g,Xji)X X i,j X dim g i j = (h[X, Xi]g, [Y, Xj]gi − h[Y, Xi]g, [X, Xj]gi)X X . i,j X (5.2.18) Comparing Equations 5.2.17 and 5.2.18 shows that Equation 5.2.16 holds, which proves Equation 5.2.15.

5.2.19 Because the Lie algebra (gˆ, [ , ]s) is isomorphic to (g, [ , ]g), the even generators ing ˆ generate the universal enveloping algebra U(g)ˆ inside W(g). Since the elements in this subalgebra commutes with the Clifford algebra Cl(g) generated by the odd generators X ∈ g, we have W(g) = Cl(g) ⊗ U(g)ˆ . (5.2.20)

80 5.2.21 The cubic Dirac operator, in terms of the new generators Xb and X, takes the following form:

dim g 1 D/ = Xi Xbi + γ(Xi) . (5.2.22) 3 i=1 X Verifying this is just a matter of following the deﬁnition; by the deﬁni- tion of Xbi, we have Xi = Xbi−γ(Xi); inserting this into Equation 5.2.10 immediately gives us Equation 5.2.22. A. Alekseev and E. Meinrenken [2, Prop. 3.4, p. 144] showed that 2 1 1 D/ = Ωb g + trg Ωb g. (5.2.23) 2 48

Here Ωb g denotes the Casimir element in U(g)ˆ , and trk Ωb g denotes the trace for the adjoint action of Ωb g on g. In terms of an orthonormal dim g dim g basis { Xi }i=1 for g, we have Ωb g = i=1 XbiXbi (Equation 2.2.16). We used the caret symbol so that Ωb g would not be confused with dim g P i=1 XiXi in {1} ⊗ U(g)¯ ⊆ W(g). Remark.P A more general form of D/ was independently introduced 2 by B. Kostant in [65], where he also calculates D/ . (We shall have a chance to see them later on; see Equation 5.4.18 and Proposi- tion 6.3.14.) The reason D/ is called the “Dirac operator” has to do with the fact that the elements of W(g) = Cl(g) ⊗ U(g)ˆ can be inter- preted as differential operators acting on the sections of the trivial bundle G × E G where E is a finite-dimensional Cl(g)-module; this is obvious for us since we already gave an identification of U(g)ˆ as the algebra D→(G) of left-invariant differential operators on G, via the algebra isomorphism 2.2.11. To recapitulate, for X in g, the element X in U(g)ˆ is identified as the directional derivative ∂ on G b Xe with respect to the left-invariant vector field Xe on G generated by X. Now, one might ask, why not identify the generator X with the directional derivative; after all, X’s also generate a copy of the universal enveloping algebra in W(g). The answer is simple; the directional derivatives should commute with the Cl(g) factor. With X identified as the directional derivative ∂ , Equation b Xe 5.2.22 then shows that the cubic Dirac operator D/ is the geometric Dirac operator (Section 6.2.6) on a vector bundle over G associated to the connection ∇ defined by 1 ∇ = ∂ + c(γ(X)), Xe Xe 3 where c denotes the Cl(g)-module structure on the vector bundle. This is not a Clifford connection (Section 6.2.3), which demands that [∇ , c(Y)] = c(∇ Y) where ∇ is the Riemannian connection on the Xe e Xe e e

81 tangent bundle of G; instead, we have 1 1 [∇ , c(Y)] = [c(γ(X)), c(Y)] = c([γ(X),Y]) X 3 3 1 2 = c([X, Y]g) = c(∇e Ye), 3 3 Xe where the last equality follows from Equation 2.2.17.

5.3 THECLASSICALWEILALGEBRA

5.3.1 Traditionally the Weil algebra of a (ﬁnite-dimensional) Lie algebra g is ∧(g)∗ ⊗ S(g∗). (It has a g-differential algebra structure which we shall review shortly.) Since we assume that g is equipped with an invariant inner product, we shall identify g∗ with g by the inner product and, thus, the Weil algebra with ∧(g) ⊗ S(g). The connection between the classical Weil algebra and the quantum Weil algebra is most obvious from Equation 5.2.20, as ∧(g) and S(g)ˆ are the associated graded algebras of Cl(g) and U(g)ˆ , respectively. To make this more precise, let S Cl (g) and S U (g)ˆ k∈Z k k∈Z k be the usual ﬁltrations for Cl(g) and U(g), respectively (see Sec- tions 4.1.26 and 5.1.10). Set

Cl(k)(g) := Clk(g), U(2`)(g)ˆ := U`(g)ˆ . Then, give W(g) the following filtration: [ W(g) = Wq(g) q∈Z where [ Wq(g) = Cl(k)(g) ⊗ U(2`)(g)ˆ . k+2`=q The associated graded algebra of W(g), which we denote by W(g) instead of gr W(g) for reasons that will be obvious in a moment, is W(g) = ∧(g) ⊗ S(g)ˆ . The generators X in g and Xb ing ˆ are of degree 1 and 2. Recall that the symbol maps for Cl(g) and U(g)ˆ , respectively, are given by the inverse of the Chevalley map 5.1.17 and the inverse of the Poincaré-Birkhoff-Witt isomorphism 4.1.12; their tensor product q−1 ⊗ PBW−1 : W(g) = Cl(g) ⊗ U(g)ˆ W(g) = ∧(g) ⊗ S(g)ˆ (5.3.2) is the symbol map for the quantum Weil→ algebra. Through this symbol map, the g-differential algebra structure on W(g) induces a g- differential algebra structure on W(g) as follows; we divide the procedure into two steps: Step 1: Recall that the Lie derivatives on W(g) were defined in terms of the even generators X = Xb + γ(X). We will need to define an analogous generator for W(g). The image of X under the symbol

82 map is q−1 ⊗ PBW−1(Xb + γ(X)) = PBW−1(Xb) + q−1γ(X) = Xb + λ(X). Here λ(X) is the image of X under the map λ : g ∧2(g) deﬁned in Deﬁnition 5.1.46. With a slight abuse of notation, we write

X := Xb + λ(X) ∈ W(g). → Clearly X is of degree 2, and it may replace Xb as a generator for W(g). We denote the set of these new generators by

g¯ := { X | X ∈ g } ⊆ W(g).

Step 2: Deﬁne the derivations ιX, LX, and d on W(g) of degree −1, 0, and 1, respectively, in such a way that the symbol map intertwines them with their counter parts on W(g) at the level of generators. that means, if we apply the symbol map q−1 ⊗ PBW−1 to Equations 5.2.6– 5.2.8, we get (keeping in mind that q−1⊗PBW−1 maps the generating subspace g ⊕ g¯ in W(g) identically to g ⊕ g¯ in W(g))

ιXY = hX, Yi, ιXY = [X, Y]g, (5.3.3)

LXY = [X, Y]g,LXY = [X, Y]g. (5.3.4) dX = X, dX = 0. (5.3.5) With these derivations, W(g) is exactly the classical Weil algebra (with the identiﬁcation g∗ ' g), which is a g-differential algebra; see [3, § 3.2].

Remark. Letg ¯ be the linear subspace spanned by the generators X = Xb + λ(X) in W(g). Since Xb and λ(X) commutes with every generator in the generating subspace g ⊕ gˆ ⊆ W(g), the new generators X also commute with every generator in the generating subspace g ⊕ g¯ ⊆ W(g). Thus, we have W(g) = ∧(g) ⊗ S(g)¯ .

5.3.6 SUPER-SYMMETRIZATION. Recall that the Chevalley map q : ∼ ∧(g) − Cl(g) is the antisymmetrization map with respect to the generators in g (see Section 5.1.15) and that the Poincaré-Birkhoff- ∼ Witt isomorphism→ PBW : S(g)ˆ − U(g)ˆ is the symmetrization map with respect to the generators ing ˆ (see Section 4.1.9). Hence, the inverse of the symbol map 5.3.→2 is the super-symmetrization map with respect to the generators X and Xb: q ⊗ PBW : W(g) = ∧g ⊗ S(g)ˆ W(g) = Cl(g) ⊗ U(g)ˆ , 1 v ··· v 7 sgn (σ) v ··· v 1 k k! s σ(1) σ(k). → σ∈S Xk (5.3.7) → N Here vi’s are vectors in g org ˆ, and sgns(σ) = (−1) where N is the number of pairs of odd vectors vi, vj with i < j such that σ(j) < σ(i). This map is a vector space isomorphism.

83 The super-symmetrization in terms of the generators X and X is also possible. Following A. Alekseev and E. Meinrenken [2], we call this map the quantization2 map and denote it by

Q : W(g) = ∧(g) ⊗ S(g)¯ W(g) = Cl(g) o U(g)¯ . (5.3.8) This is the tensor product of the Chevalley map on ∧(g) and the Poincaré-Birkhoff-Witt isomorphism→ on S(g)¯ . This is also a vector space isomorphism.

Remark. Recall that the g-differential algebra structure for the classical Weil algebra was deﬁned in terms of the generators X and X (not Xˆ) so that it resembles the g-differential algebra structure of the quantum Weil algebra. As a consequence, the super-symmetrization with respect to the generators X and X, namely the quantization map Q, intertwines the derivations ιX, LX, and d as demonstrated in [3, Lem. 4.3, p. 315]. Now let D := Q−1(D/ ) so that dD/ = dQ(D) = Q(dD). Then, by Equation 5.2.9,

[D/ , D/ ]s = Q(dD). (5.3.9) 2 The left-hand side is equal to 2D/ . For dD, we have

dim g dD = XbiXbi, (5.3.10) i=1 X dim g where { Xi }i=1 is any orthonormal basis for g [3, Prop. 5.4, p. 319]. Let us give the following notation for the element in the right-hand side of Equation 5.3.10: dim g ∆ˆg := XbiXbi. (5.3.11) i=1 X 2 At this point the word “quantization” means the inverse of the symbol map of a filtered algebra. The meaning can be made richer with the notion of a super-Poisson bracket [66]; a super-Poisson bracket on a super-commutative algebra A (in other words, the super-commutator of any two elements of A is zero) is a map { , } : A × A A such that, for homogeneous elements x, y, and z, we have (i) {x, y} = −(−1)deg(x) deg(y){y, x}, (ii) {x, {y, z}} = {{x, y}, z} + (−1)deg(x) deg(y){y, {x, z}}, (iii) the map w→7 {x, w} defines a super-derivation of degree deg(x). As A. Alekseev and E. Meinrenken point out [3, Rmk.5.5(a), p. 319], one can define a super-Poisson bracket →{ , } on W(g) in such a way that the quantization map Q intertwines the super-Poisson bracket with the super-commutator at the level of generators (which resembles very much the principle of quantization laid out by P. A. M. Dirac [30, § 21]); all we need to do is to define { , } for the generators of W(g) so that it resembles the super-commutator relations 5.2.2–5.2.4 in W(g); that is, set {X, Y} =

hX, Yi, {X, Y} = [X, Y]g, and {X, Y} = [X, Y]g. Then the super-derivations ιX, LX, and d all become inner; in particular, d(·) = {D, ·} where D is the cubic element deﬁned by Equation 5.3.14. This allows us to write Equation 5.3.9 in a more symmetric form: [D/ , D/ ]s = Q{D, D}.

84 Then, Equation 5.3.9 can be written as

2 1 D/ = Q(∆ˆ ). (5.3.12) 2 g This has an important geometrical meaning; see Theorem 7.2.7. The following lemma is from [3, Rmk.5.2(a), p. 317]; it gives an explicit expression for D:

dim g : 5.3.13 LEMMA. Let { Xi }i=1 be any orthonormal basis for g. Let D = Q−1(D/ ). Then

dim g 2 D = X X − λ(X ) , (5.3.14) i i 3 i i=1 X where λ(Xi) is the image of Xi under the map λ deﬁned in Deﬁni- tion 5.1.46.

Proof. We shall verify that the right-hand side of Equation 5.3.14 is indeed mapped to D/ under the quantization map. In other words, we wish to show that dim g dim g 2 Q X X − Q X λ(X ) = D/ . (5.3.15) i i 3 i i i=1 i=1 X X The quantization map is the super-symmetrization with respect to the generators X and X. Thus, 1 Q(X X ) = (X X + X X ). i i 2 i i i i

Since [Xi,Xi]s = [Xi,Xi]g = 0, we have XiXi = XiXi. So

Q(XiXi) = XiXi. (5.3.16) dim g Next, we calculate Q i=1 Xiλ(Xi) . By Equation 5.1.53, we have P dim g dim g − 2 Xiλ(Xi) = hXi, [Xj,Xk]giXiXjXk. (5.3.17) i=1 i,j,k=1 X X Since the above element is in the exterior algebra ∧(g) ⊗ {1} in W(g), the quantization map operates on it as the antisymmetrization map. So

g dim Q hXi, [Xj,Xk]giXiXjXk i,j,k=1 X dim g 1 = sgn(σ) hX , [X ,X ] iX X X . 3! i j k g σ(i) σ(j) σ(k) σ∈S i,j,k=1 X3 X

85 But, since sgn(σ)Xσ(i)Xσ(j)Xσ(k) = XiXjXk,

g dim Q hXi, [Xj,Xk]giXiXjXk i,j,k=1 X dim g 1 = hX , [X ,X ] iX X X . 3! i j k g i j k σ∈S i,j,k=1 X3 X dim g = hXi, [Xj,Xk]giXiXjXk. i,j,k=1 X This proves, by Equation 5.3.17, that g g dim dim Q −2 Xiλ(Xi) = hXi, [Xj,Xk]giXiXjXk. i=1 i,j,k=1 X X Then, by Equation 5.1.55,

g g dim dim Q −2 Xiλ(Xi) = −2 Xiγ(Xi). (5.3.18) i=1 i=1 X X Collecting Equations 5.2.10, 5.3.16 and 5.3.18, we have dim g dim g dim g dim g 2 2 Q X X − Q X λ(X ) = X X − X γ(X ) = D/ . i i 3 i i i i 3 i i i=1 i=1 i=1 i=1 X X X X This is our desired result Equation 5.3.15.

5.3.19 DUFLO ISOMORPHISM REVISITED. In their paper [3, Thm. 5.3, p. 318], A. Alekseev and E. Meinrenken compared the two super- symmetrization maps q ⊗ PBW and Q, and found that

Q = (q ⊗ PBW) ◦ ∂S, (5.3.20) where ∂S is an infinite order differential operator with coefficients in ∧(g) that is defined as follows. Take the power series of S(X) = j(X) ⊗ eλφ(X) (5.3.21) where sinh(ad /2) ad 4 j(X) = det1/2 X , φ(X) = 2 coth X − . adX /2 2 adX (5.3.22) View S(X) as an element of ∧(g) ⊗ S(g)∗ using the isomorphism ∗ g ' g provided by the inner product. Then ∂S is the operator we get by having the S(g)∗ factor act on {1} ⊗ S(g)ˆ as an infinite order differential operator (here we use the isomorphismg ˆ ' g) and having the ∧(g) factor act on ∧(g) ⊗ {1} by contraction (obtained by extending the contraction map ι : g End(∧(g)) to ∧(g) End(∧(g)) as an algebra homomorphism). Notice the appearance of the function j, → →

86 which is also present in the formulation of the Duflo isomorphism. A. Alekseev and E. Meinrenken used Equation 5.3.20 to prove the Du- flo isomorphism for finite-dimensional Lie algebras over R equipped with an inner product that is skew-invariant under the ad(g)-action. In what follows is a sketch of their proof. A few definitions are in order. Let (A, ι, L, d) be a g-differential algebra. An element a of A is said to be horizontal if ιXa = 0 for all X in g; it is said to be invariant if LXa = 0 for all X in g. The horizontal and the invariant elements, respectively, constitute the horizontal g subalgebra Ahor and the invariant subalgebra A of A. The basic elements of A are the ones that are both horizontal and invariant; g they constitute the basic subalgebra Abas := Ahor ∩ A of A. From the commutator relations 5.2.13–5.2.15, we see that the horizontal subalgebra of W(g) is U(g)ˆ . So the basic subalgebra of W(g) is the invariant subalgebra of U(g)ˆ . Now, by definition, LXYb = [X, Yb]s. Using the commutator relations, we find that [X, Yb]s = [\X, Y]g. Thus, the invariant subalgebra of U(g)ˆ must be the ad(g)ˆ -invariant part of U(g)ˆ , namely, the center of U(g)ˆ . Therefore, W(g)bas = Z(g)ˆ . By the same token, we have W(g) = S(g)ˆ g, where S(g)ˆ g is the ad(g)ˆ - invariant part of S(g)ˆ . ∼ Because the quantization map Q : W(g) − W(g) intertwines the contractions and the Lie derivatives, its restriction to the basic subalgebra gives a vector space isomorphism → ∼ Qg : S(g)ˆ g − Z(g)ˆ . (5.3.23) The quantization map also intertwines the differential; so the above map induces a linear isomorphism on→ the cohomologies, Qg∗ : H∗{ S(g)ˆ g } H∗{Z(g)ˆ }. This turns out to be an algebra isomorphism [3, Thm. 4.7, p. 316]. But notice that S(g)ˆ g and Z(g)ˆ , respectively,→ are in the even subspace of W(g) and W(g). Hence, H∗{S(g)ˆ g} = S(g)ˆ g, H∗{Z(g)ˆ } = Z(g)ˆ , and Qg∗ = Qg. Therefore, the vector space isomorphism 5.3.23 is an algebra isomorphism. Finally, since the elements in S(g)ˆ g have no exterior algebra factor, to the end of applying the operator 5.3.20 on S(g)ˆ g, we may drop the exterior algebra factor in the power series 5.3.21 and replace S(X) with j(X). To sum up, the algebra isomorphism 5.3.23 is equal to (q ⊗ PBW) ◦ ∂j. This is precisely the Duflo isomorphism.

5.4 THERELATIVEWEILALGEBRA

5.4.1 DEFINITION. Let (A, ι, L, d) be a g-differential algebra. Let k be a subset of g. An element a of A is said to be k-horizontal if ιXa = 0 for all X in k; it is said to be k-invariant if LXa = 0 for all X in k. The k- horizontal and the k-invariant elements, respectively, constitute the k k-horizontal subalgebra Ak-hor and the k-invariant subalgebra A of

87 A. The k-basic elements of A are the ones that are both horizontal and invariant with respect to k; they constitute the k-basic subalgebra k Ak-bas := Ak-hor ∩ A of A. In the special case where k = g, we shall omit the qualiﬁer k from the above terminologies.

5.4.2 Let k be any Lie subalgebra of g, so that

[k, k]g ⊆ k. (5.4.3) Let g = k ⊕ p be an orthogonal decomposition. Owing to the invariance of the inner product, we have

h[k, p]g, ki = −hp, [k, k]gi = 0. Hence, [k, p]g ⊆ p. (5.4.4)

5.4.5 Following A. Alekseev and E. Meinrenken [3], we call the k- basic subalgebra of W(g) as the relative (quantum) Weil algebra for the pair (g, k) and denote it by W(g, k). The contraction ιX on W(g) is the inner derivation with respect to X; so the super-commutator relations [X, Y]s = [X, Y]g and [X, Yb]s = 0 tells us that the k-horizontal subalgebra is W(g)k-hor = Cl(p) ⊗ U(g)ˆ . Hence, W(g, k) = (Cl(p) ⊗ U(g))ˆ k. The differential on W(g, k) inherited from W(g) is again inner [3, Prop. 6.4, p. 321] with respect to the element

D/ (g, k) := D/ g − D/ k, (5.4.6) where D/ g denotes the cubic Dirac operator in W(g) and D/ k denotes the image of the cubic Dirac operator in W(k) under the canonical inclusion W(k) , W(g) induced by the generators X and X. The element D/ (g, k) is called the relative Dirac operator for the pair (g, k). → 5.4.7 DEFINITION. We denote by γg : g spin(g) ⊆ Cl(g), γk : k spin(k) ⊆ Cl(k), the Lie algebra homomorphisms→ constructed in Deﬁnition 5.1.46. We denote by → γp : k spin(p) ⊆ Cl(p)

k −ad so(p) the Lie algebra homomorphism→ obtained by composing with the Lie algebra isomorphism so(p) spin(p) given by the map 5.1.42. → →

88 dim k i dim k Remark. Let { Xi }i=1 and { X }i=1 be two basis for k that are dual j with respect to the inner product so that hXi,X i = δij. Likewise, dim p dim p let { Yj }j=1 be a basis for p and let { Yj }j=1 be the dual basis. Let dim g dim k dim p i dim g { Zi }i=1 = { Xi }i=1 t { Yj }j=1 ; deﬁne { Z }i=1 similarly. By Equa- tion 5.1.55, we have, for X in k, dim g 1 γg(X) = − hX, [Z ,Z ] iZiZj, (5.4.8) 2 i j g i,j=1 X 1 dim k γk(X) = − hX, [X ,X ] iXiXj, (5.4.9) 2 i j g i,j=1 X dim p 1 γp(X) = − hX, [Y ,Y ] iYiYj. (5.4.10) 2 i j g i,j=1 X dim k dim p 5.4.11 LEMMA. Let { Xi }i=1 and { Yj }j=1 be orthonormal basis for k dim g dim k dim p and p, respectively; and let { Zi }i= = { Xi }i=1 t { Yj }j=1 . In the following, let X be an arbitrary vector in k.

(a) We have γg(X) = γk(X) + γp(X). (5.4.12)

(b) Under the canonical inclusion W(k) , W(g), we have

Xb 7 Xb + γp(X). (5.4.13) → Proof. (a) Let us rewrite Equation as follows: → 5.4.8 1 dim k dim k γg(X) = had X ,X iX X 2 X i j i j i=1 j=i X X x dim k dim p | {z } + hadX Xi,YjiXiYj i=1 j=1 X X y dim p dim p | 1 {z } + had Y ,Y iY Y . (5.4.14) 2 X i j i j i=1 j=i X X z Now, owing to the invariance of| the inner{z product, the} orthogonal decomposition g = k ⊕ p is invariant under ad(k)-action. As a consequence, the sums x, y, and z above are equal to γk(X), 0, and γp(X), respectively. (b) The generator Xb in W(k) is, by deﬁnition, equal to X − γk(X). So, under the canonical inclusion W(k) , W(g), Xb = X − γk(X) 7 X − γk(X). → → 89 Since Xb = X−γg(X) in W(g), the image above is equal to Xˆ+γg(X)− γk(X). By Equation 5.4.12, this is equal to Xb + γp(X).

5.4.15 NOTATION. The restriction of the canonical inclusion 5.4.13 onto {1} ⊗ U(ˆk) in W(k) gives us a diagonal embedding of U(ˆk) into W(g). We shall denote this embedding by diag : U(ˆk) , W(g, k), W (5.4.16) Xb 7 Xb + γp(X). → 5.4.17 The signiﬁcance of the relative→ Dirac operator comes from the following result of B. Kostant [65, Thm. 2.13, p. 474]:

2 1 1 1 1 D/ (g, k) = Ωb g − diag Ωb k + trg Ωb g − trk Ωb k. (5.4.18) 2 2 W 48 48

Here Ωb g denotes the Casimir element in U(g)ˆ , and trg Ωb g denotes the trace of its representation ong ˆ via the adjoint representation. dim g In terms of an orthonormal basis { Xi }i=1 for g, we have Ωb g = dim g i=1 XbiXbi (Equation 2.2.16). We used the caret symbol so that Ωb g would not be confused with the element dim g X X in {1} ⊗ U(g)¯ in P i=1 i i W(g). B. Kostant also demonstrated a simple formula [65, Eq. 1.85] for the trace of the Casimir element thatP holds when the Lie algebra g is compact (or reductive):

1 2 trg Ωb g = −kρgk . (5.4.19) 24

Here, ρg is half the sum of the positive roots of g, and kρgk is its norm induced by the inner product on g. This formula generalizes the “strange formula” of H. Freudenthal and H. de Vries [42, pp. 224 and 243] for the special case where g is semisimple and the inner product is coming from the Killing form.

5.4.20 What we have stated so far have their counter part in the classical Weil algebra. The classical relative Weil algebra W(g, k) is deﬁned to be the k-basic subalgebra of W(g). Similar argument used for W(g, k) shows that W(g, k) = (∧(p) ⊗ S(g))ˆ k. We set D(g, k) := Dg − Dk, (5.4.21) where Dg is the cubic element 5.3.14 for W(g), and Dk is the image of the cubic element in W(k) under the canonical inclusion W(k) , W(g) in terms of the generators X and X. The canonical inclusion W(k) , W(g) induces the diagonal em-→ bedding diag : Sˆk , W(g, k), W → (5.4.22) Xb 7 Xb + λp(X), → →

90 where λp is the composition of the map γp with the symbol map Cl(p) ∧p. By Equations 5.3.12 and 5.4.18, we have → 1 D/ (g, k)2 = Q∆ˆ − diag ∆ˆ ). (5.4.23) 2 g W k Remark (Vogan Conjecture). Because the quantization map Q : W(g) W(g) intertwines the derivations ιX, LX, and d, the restriction of Q to the k-basic subalgebra gives a vector space isomorphism → Qk : W(g, k) −∼ W(g, k). The induced map on the cohomologies, → Qk∗ : H∗{W(g, k)} H∗{W(g, k)}, is also a vector space isomorphism. A. Alekseev and E. Meinrenken showed that this is in fact an algebra→ isomorphism [3, Thm. 6.5, p. 322], and used it to prove the Vogan conjecture. In the special case where k = g, we have W(g, g) = S(g)ˆ g, W(g, g) = (U(g))ˆ g = Z(g)ˆ . This is the situation we examined in Section 5.3.19, where we saw that the induced map Qg∗ is the Duﬂo isomorphism.

91 EQUIVARIANT6 DIFFERENTIALOPERATORSAND THEQUANTUMWEILALGEBRA

UR goal of this chapter is to show that the relative Weil algebra O W(g, k) serves as an algebraic model for the space of certain G- equivariant differential operators on some vector bundle over the homogeneous space G/K, where G is a connected Lie group endowed with a bi-invariant metric and K a closed connected subgroup of G. We have hinted in a previous remark (see page 81) how this ought to be carried out. We begin by reviewing some basic notions surrounding principal bundles and associated vector bundles. Clifford module bundles and spin structures are discussed next. We then describe a natural procedure for constructing a G-equivariant Clifford module bundle over G/K, provided that G/K has spin structure. Finally, we show how an element of the relative Weil algebra can be identiﬁed as a G-equivariant differential operator acting on the sections of an equivariant Clifford module bundle. For background materials and general reference, we refer to [85, Ch.2–4].

6.1 PRINCIPALBUNDLESAND ASSOCIATEDVECTORBUNDLES

6.1.1 Let G be any Lie group. Let P be a principal G-bundle over a manifold M, and assume that G acts on P on the right. For any left G-vector space E, the Borel mixing space P ×G E is the orbit space of the G-action on P × E given by (p, v) · g = (p · g, g−1 · v) for g in G and (p, v) in P × E. If E comes from a ﬁnite-dimensional representation ν : G Aut(E), then the mixing space P ×G E is a vector bundle over M →

92 whose ﬁbers are isomorphic to E (see [63, Prop. 5.4, p. 55]). This bundle is called the vector bundle associated to P by the representation ν. If there is a need to specify the representation, then we shall write P ×ν E for P ×G E.

6.1.2 NOTATION. Let P and E be as in Section 6.1.1. For (p, v) in P × E, we shall denote its G-orbit by [p, v]. Thus, [p · g, g−1 · v] = [p, v] for all g in G.

Example (Frame Bundles). Let F M be a vector bundle of rank n over R. An ordered basis (e1, . . . , en) for the fiber Fx is called a frame. Let Fr(F) be the set of all→ frames of F. There is a canonical projection Fr(F) M, whose fiber Fr(F)x over x in M is the set of all frames for Fx. Since the general linear group GL(n, R) acts freely and transitively on→ each fiber, Fr(F) is a principal GL(n, R)-bundle. (For details on the local trivializations, see [63, Ex. 5.2, p. 55].) n With the canonical action of GL(n, R) on R , we can construct n the associated vector bundle Fr(F) ×GL(n,R) R over M. A point in n the fiber Fr(F) ×GL(n,R) R over x in M is an equivalence class [f, v] where f is a frame (e1, . . . , en) for Fx and v is an n-tuple (v1, . . . , vn) n n in R . The mapping [f, v] 7 i=1 viei gives a well-defined bundle ∼ isomorphism Fr(F) × n − F. Thus, every vector bundle is an GL(n,R) R P associated vector bundle to→ a principal bundle. If the bundle F is Riemannian→ (that is, an inner product is as- signed for each fiber in a smooth manner) then we can construct the principal O(n)-bundle FrO(F) of orthonormal frames of F. If the Rie- mannian bundle F is orientable, then we can construct the principal SO(n)-bundle FrSO(F) of oriented orthonormal frames of F.

Remark. A smooth section of Fr(F) is called a framing of F. Not all vector bundles admit a global framing; however, every vector bundle admits local framings.

6.1.3 CONNECTION. For X in g, we denote by Xe the vector ﬁeld on P whose value at p in P is deﬁned by

d Xepf = f(p · exp(tX)), (6.1.4) dt 0 where f is any smooth functions on P. This vector field is called the fundamental vector field on P generated by X. A tangent vector Y at p in P is said to be vertical if there is some X in g such that Xep = Y; the vertical vectors at p span the vertical subspace VpP of the tangent space TpP. A vector field on P is said to be vertical if it is a section of : F the subbundle VP = p∈P VpP of the tangent bundle TP. A connection on a principal G-bundle P is a g-valued 1-form θ on

93 P (that is, an element of Ω1(P) ⊗ g), such that θ(Xe) = X, ∀X ∈ g, (6.1.5) ∗ Adg ◦ θ = rg−1 θ, ∀g ∈ G, (6.1.6) ∗ where rg−1 denotes the pullback along the right-action of g on P, and Adg acts on the g-factor of the g-valued form. In regard to the properties 6.1.5 and 6.1.6, we say that the connection is vertical and invariant (or equivariant), respectively. A connection θ determines, at each point p in P, a complemen- tary subspace HpP of the vertical subspace VpP in TpP, namely,

HpP := ker(θp). This subspace is called the horizontal subspace at p with respect to θ. Tangent vectors in HpP are said to be horizontal, and so are the : F sections of the subbundle HP = p∈P HpP of TP.

Example (Maurer-Cartan Connection on Lie Groups). Let P = G and view dim g it as the principal G-bundle G pt. Let { Xi }i=1 be a basis for g. dim g ∗ Let { αi }i=1 be the dual basis for g . Then dim→g θe := αi ⊗ Xi i=1 X defines a g-valued form on the tangent space TeG. Extend this as a g-valued differential form θ on P by defining the value of θ at g ∈ G as dim g ∗ ∗ θg = Adg−1 ◦rg−1 θe = rg−1 αi ⊗ Adg−1 (Xi). (6.1.7) i=1 X This defines a connection, known as the Maurer-Cartan connection, for the principal G-bundle G pt. This example can be used to show that any principal G-bundle P admits a connection. To that→ end, we may assume, owing to local triviality, that P = M × G. Now take the Maurer-Cartan connection on the principal G-bundle G pt, and take its pullback along the projection M × G G, (m, k) 7 k. → 6.1.8 BASIC FORMS.→ Let ν : G →Aut(E) be a finite-dimensional representation. An E-valued differential form η on P (that is, an element of Ω(P) ⊗ E) is said to be horizontal→ if the interior product ιYη vanishes for all vertical vector fields Y on P; it is said to be invariant (or −1 ∗ equivariant) if ν(g ) ◦ η = rgη for all g in G. We say that η is basic if it is horizontal and invariant. The significance of basic forms is that they are the forms that can be identified with differential forms on the base space M that takes values in the associated vector bundle P ×ν E. To be more precise, a differential k-form ω on M is said to have values in P ×ν E if ω(X1,...,Xk) is a section of P ×ν E for any vector fields X1,...,Xk

94 on M. We denote by Ω(M; P ×ν E) the space of differential forms on M with values in P ×ν E. Let Ωbas(P; E) be the space of basic differential forms on P with values in E. Then, the pullback along the projection P M gives a isomorphism of graded spaces ∗ ∼ ∗ Ω (M; P ×ν E) − Ω (P; E). (6.1.9) → bas For a proof, see [63, Ex. 5.2, p. 76]. In the special case where ν : G Aut(R) is the trivial 1-dimensional→ representation, we get ∗ ∗ Ω (M) ' Ωbas(P). (6.1.10) → 6.1.11 COVARIANT DERIVATIVE. The usual exterior derivative on differential forms do not preserve the space of basic forms. The invariance is not the issue since the exterior derivative is functorial; rather, it is the horizontal property that the exterior derivative can destroy. To remedy this, deﬁne the (exterior) covariant derivative on η in Ω(P; E) by Dη := (dη) ◦ h, where d is the usual exterior derivative on Ω(P) and h : TP HP is the projection onto the horizontal subspace. The covariant derivative clearly preserves the subspace Ωbas(P; E) of basic forms. → Remark. By the linear isomorphism 6.1.9, the covariant derivative induces a differential ∇ on the (P ×ν E)-valued forms on M so that we would have the following commutative diagram:

Ω∗ (P; E) D / Ω∗+1(P; E) bas O bas O ∼ (6.1.9) (6.1.9) ∼

∗ ∇ ∗+1 Ω (M; P ×ν E) / Ω (M; P ×ν E)

We call ∇ the (exterior) covariant derivative on P ×ν E induced by the 0 connection θ on P. Now the homogenous component Ω (M; P ×ν E) ∗ of Ω (M; P ×ν E) is simply the space Γ(P ×ν E) of sections of P ×ν E. The restriction of ∇ to this subspace yields a linear map, ∇ : Γ(P × E) Ω1(M; P × E), ν ν (6.1.12) σ 7 ∇σ, → such that, if we write (∇σ)(X) =: ∇Xσ for vector ﬁelds X on M, then → (i) ∇X+φYσ = ∇Xσ + φ∇Yσ

(ii) ∇X(φσ) = (Xφ)σ + φ∇Xσ for any φ in C (M). This deﬁnes a connection for the vector bundle

P ×ν E in the conventional∞ sense.

6.1.13 CURVATURE. Of particular interest is the covariant derivative of a connection θ on P: Θ := Dθ.

95 This is called the curvature of θ. E. Cartan’s so-called (second) structural equation (see [63, Thm. 5.2, p. 77]) states that 1 Θ = dθ + [θ, θ] . (6.1.14) 2 g The bracket in this equation is the commutator for Lie-algebra valued differential forms, which is defined as follows: Suppose ω ∈ Ωk(P)⊗ ` dim g g and τ ∈ Ω (P) ⊗ g. Choose a basis { Xi }i=1 for g so that we may dim g dim g write ω = i=1 ωi ⊗ Xi, τ = i=1 τi ⊗ Xi. Then, P dim g P k+` [ω, τ]g := (ωi ∧ ωi) ⊗ [Xi,Xj]g ∈ Ω (P) ⊗ g. i,j=1 X If Y and Z are vector fields on P, then the structural equation 6.1.14 says that Θ(Y, Z) = dθ(Y, Z) + [θ(Y), θ(Z)]g, where [ , ]g is now the Lie bracket in g. The structural equation shows that the curvature form Θ is basic. In particular, it is horizontal. So Θ is completely determined by its values on the horizontal vector fields. Let Y and Z be horizontal vector fields on P. Since θ is vertical, θ(Y) = 0 = θ(Z). So the structural equation gives us Θ(Y, Z) = dθ(Y, Z). A routine calculation yields (see, for instance, [92, Prop. 2.25(f), p. 70]) dθ(Y, Z) = Yθ(Z) − Zθ(Y) − θ([Y, Z]). The first two terms on the right-hand side are zero; thus, we have Θ(Y, Z) = −θ([Y, Z]). (6.1.15)

6.1.16 CARTAN’S VIEW ON CONNECTIONS. It is worth while to mention H. Cartan’s [20] point of view on connections. We proceed in three steps: Step 1: Simplify the notation by writing ιX and LX, respectively, for the interior product ι and the Lie derivative L with respect to Xe Xe the fundamental vector ﬁeld Xe. Equation 6.1.5 can then be written as ιXθ = X. (6.1.17) For Equation 6.1.6, substitute g with exp(tX) and take the derivative with respect to t at t = 0; we get

adX ◦ θ = LXθ. (6.1.18) Step 2: Owing to the canonical isomorphism g ' (g∗)∗, the g- valued 1-form θ can be viewed as a 1-form valued linear function θ on g∗. Then Equations 6.1.17 and 6.1.18 translates as follows: For

96 any φ in g∗,

ιXθ(φ) = hX, φi, θ ◦ adˇ X = LXθ, (6.1.19) where adˇ : g End(g∗) is the dual representation of ad : g End(g), that is, hadˇ Xφ, Yi = hφ, adX Yi for Y in g. Step 3: By→ the universal property of the exterior algebra, the→ linear map θ extends to an algebra homomorphism θ : ∧(g∗) Ω(P). (6.1.20) Now set, for X in g and φ in g∗, → ιXφ := hX, φi,LXφ = adˇ Xφ. These operations extend to ∧(g∗) as graded derivations of degree −1 and 0, respectively. Then, owing to equations in 6.1.19, ιX and LX commutes with the extended map θ:

ιX ◦ θ = θ ◦ ιX,LX ◦ θ = θ ◦ LX. This is the point of view of H. Cartan on connections — a graded algebra homomorphism ∧(g∗) Ω(P) that commutes with the contractions ιX and the Lie derivatives LX. In fact, he goes on and de- ﬁnes [20, p. 21] an algebraic connection→ (connexion algébrique) as a graded algebra homomorphism from ∧(g∗) to a g-differential algebra E that commutes with ιX and LX for all X in g. The map 6.1.20 is a special case of an algebraic connection.

6.1.21 CHERN-WEIL THEORY. Historically, the classical Weil algebra W(g) = ∧(g∗) ⊗ S(g∗) was designed as a universal g-differential algebra that serves as a model space for the connections and their curvatures on principal G- bundles. More precisely, for any algebraic connection θ : ∧(g∗) E, there is a unique g-differential algebra homomorphism CW : W(g) E → that makes the following diagram commutative: → W(g) 9 inclusion CW + ∧(g∗) / E θ This was proved by H. Cartan [20]. Because CW is a homomorphism of g-differential algebras, it maps basic elements to basic elements; so we have a homomorphism CW W(g)bas −− Ebas.

→

97 This, then, induces a homomorphism of cohomologies:

∗ ∗ CW ∗ H {W(g)bas} −−− H {Ebas}. (6.1.22) Consider the case where E = Ω(P); we have → ∗ ∗ CW ∗ H {W(g)bas} −−− H {Ωbas(P)}. (6.1.23) By the isomorphism 6.1.10 and de Rham’s theorem, we have → ∗ ∗ ∗ H {Ωbas(P)} ' H {Ω(M)} ' H (M). (6.1.24) Meanwhile, as we have seen in Section 5.3.19, ∗ ∗ g H {W(g)bas} = W(g)bas = S(g ) . (6.1.25) So, by Equations 6.1.23–6.1.25, we have an algebra homomorphism S(g∗)g H∗(M). This is known as the Chern-Weil homomorphism [24,93]. The following is the pinnacle of the Chern-Weil→ theory on characteristic classes:

6.1.26 THEOREM. Let θ be a connection on a principal G-bundle P M. Let S(g∗)g H∗(M) (6.1.27→) f 7 [f(P)] be the Chern-Weil homomorphism.→ → (a) This map is independent of the connection.

(b) [f(P)] is a characteristic class of P.

(c) Let Θ be the curvature form of θ. Let f in S(g∗)g be an invariant homogeneous polynomial on g of degree n. Then f(P) is a 2n- form on M given as follows: For tangent vectors X1,...,X2n at x in M,

f(P)(X1,X2,...,X2n−1,X2n) = 1 sgn(σ)f(Θ(X¯ , X¯ ),...,Θ(X¯ , X¯ )), (2n)! σ(1) σ(2) σ(2n−1) σ(2n) σ∈S X2n (6.1.28) ¯ where Xi, 1 6 i 6 2n, are the lifts of Xi to the horizontal subspace of TpP for some ﬁxed p in the ﬁber over x.

Proof. See [76, Thm. 6.47, p. 278, Lem. 6.49, p. 281].

6.1.29 There is also a Chern-Weil homomorphism for vector bundles. A representation ν : G Aut(E) induces a Lie algebra representation ν∗ : g End(E) (see Section 2.2.5). Since the curvature Θ of a connection on P is a basic→ g-valued 2-form on P, we may apply the →

98 1 Lie algebra representation ν∗ to the g-factor of Θ; this gives us a basic End(E)-valued 2-form

ΘE := −ν∗Θ

on P. (The negative sign is necessary for ΘE to match with the usual conventions; see the discussion in [35, surrounding Eq. 6.9, p. 56– 57].) Now let Inv(E) denote the space of polynomials on End(E) that are invariant under the conjugation action of Aut(E) on End(E). Then, replacing Θ with ΘE in the formula 6.1.28 deﬁnes the Chern- Weil homomorphism for vector bundles, Inv(E) H∗(M), f 7 [f(P)]. Just like the principal bundle case,→ this map does not depend on the choice of the connection, and it maps→ each f in Inv(E) to a characteristic class of P ×ν E. We will not dwell on this any further; for details, see [85, Ch.2].

Remark. A common practice is to put an extra factor of (2πi)−1 in front of each Θ in Equation 6.1.28 to make the characteristic class ∗ [f(P)] integral, that is, [f(P)] ∈ H (M, Z). Details on how this works out, in the case of complex line bundles, can be found in [35, § 15.3].

6.2 CLIFFORDMODULEBUNDLESAND SPINMANIFOLDS

6.2.1 In this section, M is an n-dimensional manifold equipped with a Riemannian metric h , i.

6.2.2 CLIFFORD BUNDLES. Let p be a point in M. Let Cl(TpM) be the Clifford algebra (see Section 5.1.2) generated by TpM with the inner product h , ip. It is isomorphic to the Clifford algebra Cl(n) generated by n-dimensional Euclidean space. There is an action of n O(n) on Cl(n) that arises from the canonical action of O(n) on R . The associated vector bundle

Cl(M) := FrO(M) ×O(n) Cl(n) is called the Clifford bundle of M. The complex version is deﬁned as

Cl(M) := FrO(M) ×O(n) Cl(n).

If M is orientable, we may use the principal SO(n)-bundle FrSO(M) of oriented frames instead of FrO(M). Since Cl(M) is a vector bundle with ﬁbers isomorphic to Cl(n) (see the example on page 93), and since Cl(n) is isomorphic to Cl(TpM), we see that there is a bijection F between Cl(M) and the disjoint union p∈M Cl(TpM).

1 The corresponding 2-form in Ω(M; P ×G End(E)) is the curvature 2-form for the vector bundle P ×ν E with respect to the induced connection 6.1.12.

99 6.2.3 CLIFFORD MODULE BUNDLES. A Clifford module bundle is a complex vector bundle S M such that the ﬁber Sp over p in M is a module over Cl(TpM) in a smooth manner so that we have a C (M)-linear operation of→ smooth sections: Γ(Cl(M)) × Γ(S) Γ(∞S), (ψ, σ) 7 ψ · σ. Put in another way, there is a bundle morphism c : TM End(S) ' S∗ ⊗ S → → as real vector bundles, such that for any vector ﬁeld X on M, → 1 c(X) ◦ c(X) = hX, Xi1 2 where 1 is the identity map on S. The map c then extends to c : Cl(M) End(S), which is called the Clifford action of Cl(M) on S. A Clifford connection on the Clifford module bundle S is a connec- 2 tion ∇ →on S that is compatible with the Riemannian connection on M, which we also denote by ∇, in the sense that

[∇X, c(Y)] = c(∇XY) for any vector ﬁelds X and Y on M. If the Clifford bundle S is equipped with a Hermitian metric ( , ), we further require that the Clifford action and the Clifford connection satisfy the following properties.

(i) The Clifford action of tangent vectors leaves the metric invariant: (c(X)σ1, c(X)σ2) = (σ1, σ2).

(ii) The Clifford connection is a metric connection:

X(σ1, σ2) = (∇Xσ1, σ2) + (σ1, ∇Xσ2).

2 Recall that the image of ∧ (TpM) under the quantization map q : ∧TpM Cl(TpM) is the Lie algebra spin(TpM) in Cl(TpM) (see Section 5.1.40). Property (i) guarantees that the Spin(TpM)-action on S induced→ by the Clifford action is unitary. A pair (h , i, ∇) of a Hermitian metric and a compatible Clifford connection for a Clifford module bundle S is called a Dirac structure for S. Any Clifford module bundle admits a Dirac structure; if S is trivial so that S = M × E, then the existence of a Dirac structure follows from the fact that (a) the Cl(n)-module E admits an inner product that is invariant under n the action of the generators in R (see Section 5.1.44) and (b) any vector bundle with a Hermitian metric admits a metric connection (see [63, p. 118]); the existence of a Dirac structure for an arbitrary Clifford module bundle follows, owing to the existence of the partition of unity subordinate to the locally trivializing open cover for M.

2 Also known as the Levi-Civita connection. It satisﬁes ∇XhY, Zi = h∇XY, Zi + hY, ∇XZi (it is a metric connection) and ∇XY − ∇Y X = [X, Y] (torsion free). The fundamental theorem of Riemannian Geometry states that there is a unique Rie- mannian connection on a Riemannian manifold; see [22, p. 1–2].

100 What will occupy most of our discussions is the case where the Clifford module bundle splits as a vector bundle into S = S+ ⊕ S−, and the Clifford action by a vector ﬁeld X on M is an odd endomorphism that maps Γ(S+) to Γ(S−) and vice versa. Such Clifford module bundles are said to be Z/2Z-graded. If that is the case, then we demand that the Clifford connection respects the decomposition.

6.2.4 SPIN MANIFOLDS. A spin manifold is an oriented Riemannian manifold with an extra structure that provides a standard procedure to construct Clifford module bundles; that structure — called the spin structure — is the existence of a principal Spin(n)-bundle PSpin(M) and a bundle map

α : PSpin(M) FrSO(M) that is Spin(n)-equivariant in the sense that, for x in PSpin(M) and g in Spin(n), → α(g · x) = π(g) · α(x) where π is the canonical double covering Spin(n) SO(n). Note that α is a double covering when restricted to a ﬁber in PSpin(M). A manifold is orientable if and only if its 1st Stiefel-Whitney→ class 1 in H (M; Z/2Z) is zero; it admits a spin structure if and only if its 2 2nd Stiefel-Whitney class in H (M; Z/2Z) is zero; see [69, Thm. 1.2, p. 79, Thm. 1.7, p. 82]. Suppose M is a spin manifold. Then we may redeﬁne the Clifford bundle of M as

Cl(M) = PSpin(M) ×Spin(n) Cl(n).

The identification of the fiber Cl(M)p over p in M with Cl(TpM) is done similarly as in Section 6.2.2. Now suppose E is a finite- dimensional Cl(n)-module. Then the associated vector bundle

E(M) := PSpin(M) ×Spin(n) E is a Clifford module bundle. To see this, take an element of the fiber E(M)p over p, which is an equivalence class [f, v] of (f, v) in PSpin(M) × E. An element of the fiber Cl(M)p, likewise, is an equivalence class [f0, w]. One can find an element g in Spin(n) such that 0 0 −1 f · g = f. Then, [f , w] = [f, g w] in Cl(M)p; its Clifford action on −1 −1 [f, v] in E(M)p is c([f, g w])[f, v] = [f, c(g w)v]. There is a Clifford connection for E(M) induced by the Rieman- nian connection on M; here is how it goes: The tangent bundle n TM is isomorphic to the associated vector bundle FrSO(M)×SO(n) R . So there is a connection on the principal bundle FrSO(M) that corresponds to the Riemannian connection on TM. That connection on FrSO(M) then induces a connection on the Clifford module bundle PSpin(M) ×Spin(n) E. This is a Clifford connection on E(M); see

101 [69, Prop. 4.11, p. 108]. In the special case where E is the space S of n-spinors, we have the vector bundle

S(M) := PSpin(M) ×Spin(n) S. This is called the spinor bundle of M.

6.2.5 If M is even-dimensional, any Clifford module bundle is (owing to Equation 5.1.7) of the form S(M) ⊗ W where W is some auxiliary complex vector bundle on which the Clif- ford bundle Cl(M) acts trivially. Also, since the spinor space has a natural Z/2Z-grading (see Section 5.1.44), the Clifford module bundle is Z/2Z-graded.

6.2.6 DIRAC OPERATORS. Let M be a spin manifold and ∇ be the Clifford connection on the Clifford module bundle E(M). The geometric Dirac operator is the 1st order differential operator on the vector bundle E(M) which is locally expressed as n / D = c(ei)∇ei , (6.2.7) i=1 X where (e1, . . . , en) is a local orthonormal framing of the tangent bundle of M. If the connection ∇ is induced by the Riemannian connection on M, then the geometric Dirac operator is said to be Rieman- nian. The geometric Dirac operator is globally deﬁned and independent of the choice for the framing because it is equal to the following composition of operations:

h , i D/ : Γ(E(M)) −∇ Γ(T ∗M ⊗ E) −− Γ(TM ⊗ E) −c Γ(E(M)).

If the bundle E(M) is Z/2Z-graded, its space of sections is naturally graded: → → → Γ(E(M)) = Γ(E(M)+) ⊕ Γ(E(M)−). (This happens when M is even-dimensional, as pointed out in Sec- tion 6.2.5.) In this case, the Dirac operator is an odd operator because of our requirement that the Clifford action of a tangent vector be odd (see Section 6.2.3). Geometric Dirac operators belong to the more general class of Dirac operators [12, Def. 3.36, p. 116]; a Dirac operator D/ on M is a 1st-order differential operator on a vector bundle over M (an odd operator if the bundle is Z/2Z-graded) such that its square is a gen- 2 eralized Laplacian, which means that D/ is a 2nd-order differential operator whose expression in local coordinates is of the form

2 1 D/ = ηij∂ ∂ + (lower order part), 2 i j i,j X

102 ij where η is the (i, j)-entry of the inverse of the matrix [ηij] coming i j from the metric η = i,j ηijdx dx on M. P 6.2.8 SPIN STRUCTURE ON HOMOGENEOUS SPACES. We now focus our attention to the case M = G/K where G is a Lie group (not necessarily compact) and K is a closed connected subgroup. Assume that G admits a bi-invariant metric, that is, there is an Ad(G)-invariant inner product on g. Let g = k ⊕ p be an orthogonal decomposition. Let k be an arbitrary element of K. Clearly, Adk(k) ⊆ k. Then Adk(p) ⊆ p because −1 0 = hAdk (k), pi = hk, Adk(p)i. Thus, p is a K-vector space, and since K is connected, this representation maps K into SO(p). We view G as a principal K-bundle over G/K. The vector bundle associated to the representation

K −Ad SO(p) (6.2.9) is (isomorphic to) the tangent bundle of G/K: → T(G/K) ' G ×Ad p. This gives us means to identify the tangent space at any point in G/K with p. The set of orthonormal frames for p can be identiﬁed with SO(p). Let K act on SO(p) on the left via the representation 6.2.9. The associated vector bundle is (isomorphic to) the orthonormal oriented frame bundle:

FrSO(G/K) ' G ×K SO(p). Seeking a spin structure for G/K, suppose the orthogonal action of K on p can be lifted to Spin(p): Spin(p) : Ad f (6.2.10) K / SO(p) Ad

Let K act on Spin(p) on the left via Ad.f Then the mixing construction

PSpin(G/K) := G ×K Spin(p) gives a principal Spin(p)-bundle over G/K, and there is an obvious bundle map PSpin(G/K) FrSO(G/K). So we have a spin structure on G/K. Conversely, a spin structure on G/K implies the existence of the→ lift 6.2.10 if G is simply connected [11, Lem. 3, p. 66] or G is compact connected with dim(G/K) > 3 [68, p. 14].

103 6.3 EQUIVARIANTDIFFERENTIALOPERATORS ANDTHERELATIVEWEILALGEBRA

6.3.1 We continue with the assumptions made in Section 6.2.8. Let ν : K Aut(E) be a ﬁnite-dimensional representation. By the linear isomorphism 6.1.9, we have an isomorphism of graded spaces, → ∗ ∼ ∗ Ωbas(G; E) − Ω (G/K; G ×ν E). (6.3.2) At the level of degree 0, we have on the left-hand side the space of E-valued functions that are K→-invariant, and on the right-hand side the space of sections of G ×ν E. So we have a linear isomorphism ∼ K Ξ : Γ(G ×ν E) − (C (G) ⊗ E) . (6.3.3) The action of K on C (G) ⊗ E is by R ⊗∞ν where R denotes the right- → regular action. Let us∞ spell out how the isomorphism Ξ works. Let σ¯ be a section of G ×ν E. Its value at gK is an equivalence class [g, wg] for some wg in E. Had we chosen a different representative for [g, wg], say (gk, wgk), then since [gk, wgk] = [g, ν(k)wgk] for any k in K, we have [g, wg] = [g, ν(k)wgk], which implies

ν(k)wgk = wg. (6.3.4) Now deﬁne the E-valued function σ on G by

σ(g) = wg. Then,

σ(gk) = wgk −1 = ν(k) wg (by Equation 6.3.4) = ν(k)−1σ(g). Hence, σ is an element of (C (G) ⊗ E)K; this is exactly the image of

σ¯ under Ξ. ∞

6.3.5 We wish to identify an element of the relative Weil algebra (see Section 5.4 for deﬁnitions and notations) k W(g, k) := (U(g)ˆ ⊗ Cl(p)) as a differential operator on (C (G) ⊗ E)K. The most natural way to proceed is to use the algebra isomorphism∞ ∼ τ ⊗ 1 : U(g)ˆ ⊗ Cl(p) − D(G) ⊗ Cl(p) (6.3.6) where τ is the algebra isomorphism 2.2.11 (here we use the fact that gˆ is a copy of g) and 1 is the identity→ automorphism of Cl(p). This means that an element of W(g, k) acts on (C (G) ⊗ E)K as follows:

(i) The action of Cl(p)-factor on (C (G) ∞⊗ E)K is by the Clifford

module structure on E. ∞ (ii) For the action of U(g)ˆ -factor, it is sufﬁcient to know the ac-

104 tion of the generators ing ˆ; for X in g, the generator Xb acts as the directional derivative ∂X with respect to the left-invariant vector ﬁeld on G generated by X. In other words, for σ ∈ (C (G) ⊗ E)K and g ∈ G,

∞ d (Xσb )(g) = σ(g exp(tX)). (6.3.7) dt 0

Let G act on (C (G) ⊗ E)K by L ⊗ 1 where L is the left-regular

representation on∞C (G) and 1 is the trivial representation on E. K Then, the differential∞ operators on (C (G) ⊗ E) that are obtained from W(g, k) in the above manner are ∞G-equivariant. Remark. Note that there is no mention of connections in the above identiﬁcation procedure. Though it is possible to introduce a suit- able3 connection on the trivial bundle G × E G for this task, we prefer not to, keeping the use of differential geometry to the mini- mum. →

6.3.8 Let Y be a vector in p. Then, Yb in W(g, k) acts on (C (G) ⊗ E)K as a differential operator according to Equation 6.3.7.∞ We wish to deﬁne the action of Yb on Γ(G ×ν E) so that the following diagram commutes:

(C (G) ⊗ E)K Yb / (C (G) ⊗ E)K O O ∼ ∼ ∞ Ξ ∞ Ξ (6.3.9)

Γ(G ×ν E) / Γ(G ×ν E) Yb

We need to specify the value Ybσ¯ at gK, where g is an arbitrary element of G. To that end, let t 7 w(t) be a curve in E deﬁned over some interval containing 0, such that → σ¯(g exp(tY)K) = [g exp(tY), w(t)] ∈ G ×ν E. (6.3.10) This is possible because p G/K, (6.3.11) X 7 g exp(X)K, is a local diffeomorphism near→ 0 (see [54, Lem. 4.1, p. 123]). Then, we deﬁne the value of Ybσ¯ at→gK to be 0 (Ybσ¯)(gK) := [g, w (0)]. (6.3.12)

We claim that this is the action of Yb on Γ(G ×K E) that makes the diagram 6.3.9 commutative. Indeed, let σ := Ξ(σ¯). Then σ satisﬁes

3 We discussed this matter brieﬂy on pages 81 and 82. To recapitulate, if k = 0, then 1 take the 3 γ term that appears in Equation 5.2.22 as the connection 1-form for the 1 p vector bundle; if k 6= 0, then use 3 γ . For more discussions in this direction, see [1].

105 (see Section 6.3.1) σ(g exp(tY)) = w(t); hence, Yσb (g) = w0(0). This implies that the value of Ξ−1(Yσb ) at gK is Ξ−1(Yσb )(gK) = [g, w0(0)]. By Equation 6.3.12, this is equal to (Ybσ¯)(gK).

6.3.13 LEMMA. Identify p with the tangent space of G/K at gK by the differential of the map 6.3.11 at 0. For Y in p and σ¯ in Γ(G×ν E), deﬁne σ¯ 7 Ybσ¯ by Equation 6.3.12. Then Yb is the directional derivative at gK with respect to the tangent vector Y. → Proof. Let V be a neighborhood of 0 in p, and U a neighborhood of gK in G/K, such that V is mapped diffeomorphically onto U by the map 6.3.11; denote this diffeomorphism between V and U by ψ : V U. The inverse of ψ yields a coordinate chart near gK. → Denote the vector bundle projection G ×ν E G/K by ξ. Note that the construction of the curve w by Equation 6.3.10 amounts to a local trivialization → ϕ : ξ−1(U) U × E over U. 4 → Letσ ¯ϕ,ψ be the expression of the sectionσ ¯ in the chart ψ and the trivialization ϕ. Then, Equation 6.3.10 translates into

σ¯ϕ,ψ(tY) = w(t). The derivative with respect to t at t = 0 is precisely the directional derivative ofσ ¯ϕ,ψ with respect to the tangent vector Y at 0 in p: 0 (∂Yσ¯ϕ,ψ)(0) = w (0). Meanwhile, Equation 6.3.12 translates into, 0 (Ybσ¯ϕ,ψ)(0) = w (0).

Therefore, Yb = ∂Y.

6.3.14 PROPOSITION. Let Yb ∈ pˆ act on σ¯ ∈ Γ(G ×K E) as Equa- tion 6.3.12 so that we have the commutative diagram 6.3.9. Then the action of the relative Dirac operator D/ (g, k) on (C (G) ⊗ E)K agrees

4 ∞ ψ σ¯ Explicitly,σ ¯ϕ,ψ : V E is given by the series of compositions, V − U − −1 ϕ p2 ξ (U) − U × E −− E where p2 is the projection onto the second component. → → → → →

106 with the action of the following differential operator on Γ(G ×ν E): dim p dim p 1 p D/ := Ybi ⊗ Yi + 1 ⊗ Yiγ (Yi) ∈ pˆ ⊗ Cl(p), (6.3.15) g/k 3 i=1 i=1 X X dim p where { Yi }i=1 is any orthonormal basis for p.

Remark. As the proof demonstrates, D/ g/k is just another expression for the relative Dirac operator D/ (g, k). The element D/ g/k was ﬁrst introduced by B. Kostant [65]. That it is equal to D/ (g, k) was observed by A. Alekseev and E. Meinrenken [3].

Proof. Owing to our discussion in Section 6.3.8, we may take the right-hand side of Equation 6.3.15 as an element of W(g, k) and show its equality with D/ (g, k). In other words, we wish to show that

dim p dim p 1 p D/ (g, k) = YbiYi + Yiγ (Yi) (6.3.16) 3 i=1 i=1 X X in W(g, k). dim k dim p Let { Xi }i=1 and { Yj }j=1 be orthonormal basis for k and p, respectively. By Equation 5.2.22, we have

dim k dim p dim k dim p 1 g 1 g D/ g = XbiXi + YbjYj + Xiγ (Xi) + Yjγ (Yj). 3 3 i=1 j=1 i=1 j=1 X X X X

(6.3.17) g k p By Equation 5.4.12, γ (Xi) = γ (Xi) + γ (Xi). So Equation 6.3.17 can be rewritten as dim k dim p D/ g = XbiXi + YbjYj i=1 j=1 X X dim p 1 dim k 1 dim k 1 + X γk(X ) + X γp(X ) + Y γg(Y ). (6.3.18) 3 i i 3 i i 3 j j i=1 i=1 j=1 X X X g For the factor γ (Yj) in the last summation above, we have, by Equa-

107 tion 5.4.8, dim k dim k g γ (Yj) = − hYj, [Xi,Xk]gi XiXk i=1 k>i X X = 0 k dim p dim p dim p dim | {z } − hYj, [Xi,Yk]giXiYk − hYj, [Yi,Yk]giYiYk i=1 k=1 i=1 k>i X X X X p = γ (Yi) dim k dim p | {z } p = hXi, [Yj,Yk]giXiYk + γ (Yj). i=1 k=1 X X Thus, for the last summation in Equation 6.3.18, we have dim p 1 Y γg(Y ) 3 j j j=1 X dim p dim p dim p 1 dim k 1 = − hX , [Y ,Y ] iX Y Y + Y γp(Y ) 3 i j k g i j k 3 j j i=1 j=1 k=1 j=1 X X X X dim p 2 dim k 1 = X γp(X ) + Y γp(Y ). 3 i i 3 j j i=1 j=1 X X Inserting this into Equation 6.3.18, we get dim k dim k 1 k D/ g = XbiXi + Xiγ (Xi) 3 i=1 i=1 X X dim p dim k dim p g 1 p + YbjYj + Xiγ (Xi) + Yjγ (Yj). (6.3.19) 3 j=1 i=1 j=1 X X X For D/ (g, k), we must subtract from the above the image D/ k of the dim k 1 dim k k cubic Dirac operator i=1 XbiXi + 3 i=1 Xiγ (Xi) in W(k) under the canonical inclusion Wk , Wg. By Lemma 5.4.11(b), P P dim k dim k dim k 1 k p D/ k = XbiXi + →Xiγ (Xi) + Xiγ (Xi). (6.3.20) 3 i=1 i=1 i=1 X X X Subtracting this from Equation 6.3.19 gives us Equation 6.3.16.

108 THEASYMPTOTICEXPANSION7 OFTHEHEATKERNEL ASSOCIATEDTOTHE CUBICDIRACOPERATOR

ET G be a compact connected Lie group with a bi-invariant metric, L and let K be a closed connected subgroup. The metric deﬁnes an invariant inner product on the Lie algebra g of G, and the Lie algebra k of K is a Lie subalgebra of g. The pair (g, k) deﬁnes the relative Dirac operator D/ (g, k); its square is given by (owing to Equations 5.4.18 and 5.4.19)

2 1 1 1 2 1 2 D/ (g, k) = Ωb g − diag Ωb k − kρgk + kρkk . 2 2 W 2 2 See Notation 5.4.15 and Section 5.4.17 for the notations. This is an element of the relative Weil algebra W(g, k) := (U(g)ˆ ⊗ Cl(p))k. As such, it can be identified as a differential operator via the algebra isomorphism ∼ τ ⊗ 1 : U(g)ˆ ⊗ Cl(p) − D(G) ⊗ Cl(p) defined in Section 6.3.5. The image of Ωb g under τ ⊗ 1 is the Lapla- → cian on G (this is essentially Equation 2.2.18). But, because of the 1 / 2 term − 2 diagW Ωb k, the image of D (g, k) under τ ⊗ 1 is, a priori, not of Laplace type. Still, we shall see that, as an operator on the do- K main (C (G) ⊗ E) (where E is a finite-dimensional Cl(p)-module), / 2 D(g, k) ∞is equal to a generalized Laplacian. So it makes sense to consider the heat kernel of (τ ⊗ 1)(D/ (g, k)2). Its asymptotic expansion, then, follows essentially from that of the scalar Laplacian on G (which we calculated in Theorem 4.2.37). To avoid the cluttering of notations, we shall suppress τ ⊗ 1 in

109 our discussions; it will be clear from the context whether we are considering an element of W(g, k) as is, or as its image under τ ⊗ 1. Similarly, we shall identify elements in W(g, k) = (S(g)ˆ ⊗∧(p))k with elements in D(g) ⊗ ∧(p).

7.1 THECONVOLUTIONKERNELOF AGENERALIZEDLAPLACIAN

As a Dirac operator D/ is a “square root” of a generalized Laplacian, by the heat kernel associated to D/ we mean the heat kernel of the 2 generalized Laplacian D/ . In this section, we briefly review some basic facts regarding generalized Laplacians and their heat kernels. It will be a slight generalization of what we saw in Section 2.1.9. Let F be a vector bundle over a compact oriented Riemannian manifold M. We assume that F is equipped with a fiber-wise inner product in a smooth manner. A generalized Laplacian on Γ(F) is a differential operator L whose expression in local coordinates takes the form 1 dim M L = ηij∂ ∂ + (lower order part), (7.1.1) 2 i j i,j=1 X ij where the η is the (i, j)-entry of the inverse of the matrix [ηij] com- i j ing from the metric η = i,j ηijdx dx on M. The one-half factor above is an insignificant part in our discussion; we put it here on P account of Equation 5.4.18. tL The heat kernel Qt of L is the integral kernel of the operator e (t > 0), that is,

tL (e σ)(x) = Qt(x, y)σ(y) voly ZM for any section σ of F. Here voly is the Riemannian volume form at y ∗ : ∗ ∗ ∗ in M. Thus, Qt is a smooth section of the bundle FF = p1F⊗p2F , where p1 and p2, respectively, are the projections M × M M onto the ﬁrst and second components. For existence of the smooth kernel Qt, see [85, Prop. 5.31, p. 83]. The heat kernel Qt has the→ following fundamental properties:

(i) It satisﬁes the heat equation:

(∂t − L)Qt(x, y) = 0, where L applies to the variable x.

(ii) It converges to the Dirac delta distribution δ(x − y) as t 0, in the sense that, for any smooth section σ of F, → Qt(x, y)σ(y) voly σ(x) ZM uniformly. →

110 The smooth kernel Qt is uniquely determined by the above two properties; see [85, Prop. 7.5, p. 96]. Suppose a compact Lie group G acts transitively on the manifold M as isometries. Then we have the identiﬁcation M ' G/K where K is the stabilizer of a point x0 in M of our choice. Let us write x¯ := xK ∈ G/K. Assume that the bundle F is homogeneous over G/K, in the sense that F = G ×K V for some ﬁnite-dimensional K-vector space V. There is a natural G-action on F given by 0 0 g · [g, v] := [g g, v] 0 for g in G and [g, v] in G×K V. Denote this action by α : G Aut(F). This induces a G-action on the space Γ(F) of smooth sections of F: −1 → (g · σ)(x¯) := αg(σ(g x¯)). Here g−1x¯ denotes g−1xK. Suppose the generalized Laplacian L on Γ(F) is equivariant with respect to this G-action. Then, for any section σ of F, we have etL(g · σ) = g · (etLσ). This means, in terms of the heat kernel,

−1 −1 Qt(x,¯ y¯)αgσ(g y¯) voly¯ = αgQt(g x,¯ y¯)σ(y¯) voly¯. (7.1.2) ZM ZM Now the Riemannian volume form on M is G-invariant, provided that the metric on M is G-invariant. If this is the case, Equation 7.1.2 gives

−1 Qt(x,¯ gy¯)αgσ(y¯) voly¯ = αgQt(g x,¯ y¯)σ(y¯) voly¯. (7.1.3) ZM ZM This implies −1 Qt(x,¯ gy¯)αg = αgQt(g x,¯ y¯), or equivalently (by takingx ¯ = e¯ and g = x−1), −1 −1 Qt(e,¯ x y¯) = αx Qt(x,¯ y¯)αx. (7.1.4)

Hence, the heat kernel Qt is completely determined by the function

qt : y¯ 7 Qt(e,¯ y¯).

We shall call qt the heat convolution kernel of L. (If K = {e} and F → is the trivial line bundle over G, then qt is the heat convolution kernel 2.1.11 we deﬁned in Section 2.1.9.) We may view it as a function ∗ ]0, [ Γ(Fe¯ F ), t 7 qt. For small time t, the heat convolution

∞ → → 111 kernel behaves like the Gaussian1 kernel e−d(x¯)2/2t h (x¯) = , (7.1.5) t (2πt)dim M/2

where d(x¯) denotes the distance betweene ¯ andx ¯. More precisely, qt admits an asymptotic expansion (or an asymptotic heat kernel) 2 qt ∼ ht(a0 + a1t + a2t + ··· ) (7.1.6) 2 r ∗ for t 0+, valid in the Banach space of C -sections of Fe¯ F for all non-negative integer r (see [85, Thm. 7.15, p. 101]). As→ it was for the asymptotic expansion for the heat kernel of the scalar Laplacian on G (see Section 2.1.9), the asymptotic expansion 7.1.6 is obtained as the formal solution to the heat equation equation

i (∂t + L)ht ait = 0, (7.1.7) ∞ i=0 X under the condition that a0(e¯) is the identity operator on the ﬁber Fe¯. Here is a sketch of the argument: Let ∇ be the connection on the vector bundle F that is associated3 to L. Let R be the radial vector ﬁeld in normal coordinates. Let ˜ := j ◦ log, where j is the square-root of the Jacobian of the Riemannian exponential map ate ¯. Then, Equation 7.1.7 is equivalent to (see [85, Eq. 7.16, p. 102] or [12, Prop. 2.24, p. 82])

1 1 i ht ∂t + ∇R + ˜ ◦ L ◦ ait = 0. (7.1.8) t ˜ ∞ i=0 X Setting each coefﬁcient of the above power series in t to zero, we get the following family of differential equations that can be solved inductively:

∇Ra0 = 0, a (∇ + n)a = −˜L n−1 , n 1. R n ˜ > For details, see [85, Thm. 7.15, p. 101] or [12, Thm. 2.26, p. 83, Thm. 2.29, p. 85].

7.2 THEASYMPTOTICEXPANSIONOF THEHEATKERNELOF D/ (g, k)2

1 The difference between Equations 2.1.16 and 7.1.5 owes to the factor of 1/2 present in our deﬁnition (Equation 7.1.5) of a generalized Laplacian. 2 The norm is deﬁned similarly as in Footnote 2 on page 16. 3 For any generalized Laplacian L on a vector bundle F, there is a connection ∇ F on F such that L is equal to (up to a zero-order differential operator) ∆ := dim M dim M − (∇e ∇e − ∇∇ e ) where { ei } is a local orthonormal framing of i=1 i i ei i i=1 TM. See [12, Prop. 2.5, p. 66]. P

112 7.2.1 SET UP. Let G be a compact connected Lie group with a bi- invariant metric. Let K be a closed connected subgroup of G, such that G/K admits a spin structure. Let p be the orthogonal complement of k in g. As we discussed in Section 6.2.8, the bi-invariance of the metric implies that the orthogonal decomposition g = k ⊕ p is preserved under the action of Adk for all k in K, and that the restriction of Adk to p is in SO(p). The spin condition on G/K implies that this representation of K on p lifts to Spin(p). Spin(p) : Adf K / SO(p) Ad

The induced Lie algebra homomorphism of Adf was considered in Deﬁnition 5.4.7, and we gave it the notation γp : k spin(p). So we have the commutative diagram → K Adf / Spin(p) O O exp exp

k / spin(p) γp

We consider ﬁnite-dimensional Cl(p)-modules of the form E = S ⊗ V where S is the space of spinors for Cl(p), and V is an irreducible K- vector space with highest weight µ, serving as an auxiliary space on which Cl(p) acts trivially. We denote the representation of K on V by π : K Aut(V). Then, K acts on E via ν :=→Adf ⊗ π. The induced Lie algebra representation of ν is the differential of ν at the identity: p ν∗ = Adf ∗ ⊗ 1 + 1 ⊗ π∗ = γ + π∗. (7.2.2) p Each Lie algebra representation ν∗, γ , and π∗ extends to U(ˆk) as an algebra homomorphism.

7.2.3 LEMMA. As differential operators on (C (G) ⊗ E)K, we have

2 ∞ 2 diagW Ωb k = −kµ + ρkk + kρkk .

Proof. Let σ be any element of (C (G) ⊗ E)K. By the K-invariance of

∞

113 σ, we have σ(g exp(X)) = e−ν∗(X)σ(g) for g in G and X in k. Differentiating both sides with respect to X, we get

Xσb = −ν∗(X)σ. By Equation 7.2.2, we may rewrite this as p (Xb + γ (X))σ = −π∗(X)σ. p Since Xb + γ (X) = diagW Xb by deﬁnition (see Notation 5.4.15), we have diagW(Xb)σ = −π∗(X)σ. Therefore, dim k dim k

diagW(Xbi) diagW(Xbi)σ = π∗(Xi)π∗(Xi)σ. i=1 i=1 X X Because diagW and π∗ are algebra homomorphisms, we have dim k dim k diagW XbiXbi σ = π∗ XiXi σ. i=1 i=1 X X The left-hand side is diagW(Ωb k)σ and the right-hand side is π∗(Ωk)σ. Hence, we have diagW(Ωb k) = π∗(Ωk) as differential operators on (C (G) ⊗ E)K. Finally, owing to the irreducibility of π and Schur’s lemma,∞ π∗(Ωk) is a constant operator, whose value is 2 2 −kµ + ρkk + kρkk ; see [65, Rmk. 1.89, p. 469]. This proves the lemma.

7.2.4 THEOREM. As a differential operator on (C (G) ⊗ E)K, the ele- / 2 ment D(g, k) is equal to ∞ 1 1 2 2 Ωb g + kµ + ρkk − kρgk . (7.2.5) 2 2 Proof. By Equations 5.4.18 and 5.4.19,

2 1 1 1 2 1 2 D/ (g, k) = Ωb g − diag Ωb k − kρgk + kρkk . 2 2 W 2 2 The theorem now follows from Lemma 7.2.3.

7.2.6 In the following, ∆ˆg is the element in the classical Weil algebra deﬁned by Equation 5.3.11. It actually lies in the basic subalgebra S(g)ˆ g. It is identiﬁed with the Laplacian on the Euclidean space g under the algebra isomorphism 4.1.4 with A = g.

114 7.2.7 THEOREM. Let Q be the quantization map 5.3.8 (restricted to the k-basic subalgebra). Then 2 Q(∆ˆg) = Ωb g − kρgk .

Proof. By Equation 5.3.12, 1 Q(∆ˆ ) = D/ 2, 2 g g where D/ g is the cubic Dirac operator of W(g). Meanwhile, by Equa- tions 5.2.23 and 5.4.19,

2 1 1 2 D/ = Ωb g − kρgk . g 2 2 Combining the two equations above proves the theorem.

−1 7.2.8 DEFINITION. Let expg∗ : D(G) D(g) be the map 4.2.16, and let q : ∧(p) Cl(p) be the Chevalley map 5.1.17. For a differential operator L in D(G) ⊗ Cl(p), we set → → exp −1 −1 L := (expg∗ ⊗ q )(L). Remark. (1) We have a vector space isomorphism, D(G) ⊗ Cl(p) D(g) ⊗ ∧(p), (7.2.9) L 7 Lexp. → (2) Let σ := f⊗w be an element of C (G)⊗E. Let σexp := fexp ⊗w, exp → where f is deﬁned by Equation∞4.2.14. Let the D(g)-factor of Lexp act on f, and let the ∧(p)-factor act on w via the Chevalley map. Then, by Equations 4.2.15 and 4.2.18, Lexpσexp = (Lσ)exp. In this sense, we may understand Lexp as the expression for L under a local exponential chart near the identity in G.

7.2.10 Consider, for the moment, the case where k = g. The corresponding (classical) relative Weil algebra is W(g, g) = S(g)ˆ g. The element ∆ˆg lies in this “smallest” relative Weil algebra; and the restriction of the quantization map Q to this subalgebra, as we have seen in Section 5.4.20, is the Duﬂo isomorphism. So it is not surpris- ing to see the j-factor appearing in the following Proposition.

7.2.11 PROPOSITION. Let j be the function on g deﬁned by the power series sinh ad /2 j(X) = det1/2 X . adX /2 Then, 1 (Q(∆ˆ ))exp = ◦ ∆ˆ ◦ j. g j g

115 Remark. The above equation is that of differential operators on g; so j and 1/j are to be understood as the multiplication by the functions j and 1/j, respectively.

g Proof. The element ∆ˆg lies in S(g)ˆ , which is the basic subalgebra of the classical Weil algebra W(g); the quantization map Q restricted this subalgebra is the Duﬂo isomorphism (see Section 5.3.19). The assertion now follows from Proposition 4.2.20.

7.2.12 NOTATION. For the rest of the Chapter, qt denotes the heat convolution kernel of Q(∆ˆg)/2. Then qt is an element of C (G) ⊗ exp exp Cl(p). We deﬁne qt similarly as we have done for σ associated ∞ g to σ in C (G) ⊗ E (Remark (2) on page 115). We shall denote by ht ˆ the heat∞ convolution kernel of ∆g/2, that is, −kXk2/2t g e h (X) := . (7.2.13) t (2πt)dim g/2

7.2.14 PROPOSITION. For t 0+, we have 1 qexp ∼ hg · , →t t j valid in some neighborhood of 0 in g. In other words, the coefficients in the expansion 7.1.6, for this case, are a0(x) = 1/j(log(x)) and an(x) = 0 for n > 1. g exp : i Proof. Let st = ht i=0 ait be the asymptotic expansion for qt (or its product with a bump∞ function ψ such that ψ ≡ 1 near 0). It is the formal solution toP 1 ∂ + Q(∆ )exp s = 0. t 2 g t g We need to show that st = ht /j satisfies this equation. By Proposi- tion 7.2.11, the above is equivalent to 1 1 ∂ − ∆ˆ js = 0. j t 2 g t g g It is now clear that st = ht /j is the solution, since ht satisfies the 1 g heat equation (∂t − 2 ∆g)ht = 0.

7.2.15 THEOREM. Let rt be the heat convolution kernel of the generalized Laplacian 1 1 2 2 Ωb g + kµ + ρkk − kρgk 2 2 K on (C (G) ⊗ S ⊗ V) , where S is the spinor space for Cl(p) and V is exp an irreducible∞ K-vector space with highest weight µ. Then rt has the following asymptotic expansion, valid in a neighborhood of 0 in g: For t 0+, 2n 1 2 1 kρ + µk exp g tkρk+µk /2 g k n → rt ∼ ht · e = ht n t . j ∞ j 2 · n! n=0 X

116 Proof. By Theorem 7.2.7,

1 1 2 2 1 1 2 Ωb g + kµ + ρkk − kρgk = Q(∆ˆg) + kρk + µk . 2 2 2 2 So 2 2 ˆ 2 et(Ωb g+kµ+ρkk −kρgk )/2 = etQ(∆g)/2etkρk+µk /2. Hence, 2 tkρk+µk /2 rt = e qt, where qt is the heat convolution kernel of Q(∆ˆg)/2. The asymptotic exp expansion for rt now follows from Proposition 7.2.14.

Remark. Let qt be the heat convolution kernel of Q(∆ˆg)/2. Deﬁne tQ∆ˆ /2 K the linear map (e g )e :(C (G) ⊗ E) E by

∞ tQ∆ˆg/2 (e )eσ = qt(x)σ→(x) dx. ZG t∆ˆ /2 K Deﬁne also the linear map Q(e g )e :(C (G) ⊗ E) E by

∞ t∆ˆg/2 g Q(e )eσ = j(log x)ht (log x)σ(x)ψ(x) dx,→ ZG where ψ is a suitable bump function on G such that

(i) its support is contained in some neighborhood of the identity on which the inverse of the exponential map log = exp−1 is well-deﬁned,

(ii) ψ(x) = 1 for x near the identity.

Then Proposition 7.2.14 is equivalent to the following asymptotic equality: t∆ˆg/2 tQ∆ˆg/2 Q(e )eσ ∼ (e )eσ for t 0+. This asymptotic equality can also be proved using the (now proved) Kashiwara-Vergne conjecture [4, 60]. →

117 THELOCALINDEXTHEOREM8 ONCOMPACT HOMOGENEOUSSPACES

E now come to the primary matter — the local index theo- W rem on a compact homogeneous space G/K. We obtained, in Chapter 7, the asymptotic expansion for the heat kernel associated to the relative Dirac operator D/ (g, k) using the machinery of quantum Weil algebra. Now the operator D/ (g, k) is a differential operator on (C (G) ⊗ E)K some Clifford module E; the corresponding / differential operator∞ on Γ(G ×K E) is the cubic Dirac operator Dg/k (Proposition 6.3.14). Our goal is to deduce the asymptotic heat ker- 2 2 nel of D/ g/k from that of D/ (g, k), and achieve the local index theorem on G/K. Our approach is similar to N. Berline and M. Vergne’s [13] proof of the local index theorem. They considered the Riemannian Dirac operator D/ on the vector bundle PSpin ×Spin(n)E M, where M is a generic spin manifold of dimension n and PSpin is the principal Spin(n)-bundle that is a double covering of the bundle→ FrSO(M) of oriented orthonormal frames for the tangent bundle TM (see Sec- tion 6.2.4). They showed that the square of the Dirac operator D/ can be identiﬁed with a second order differential operator Q on (C (P) ⊗ E)Spin(n) and that the operator Q is equal (up to a zero- ˜ degree∞ operator) to a generalized Laplacian ∆. They calculated the asymptotic heat kernel of ∆˜ and, from it, deduced the asymptotic 2 heat kernel of D/ on the base manifold M. Our approach, in outline, follows the method of N. Berline and M. Vergne. What differs is that, for M = G/K, we use

(i) the K-action on the spinors, instead of the full spin group action,

118 (ii) the natural principal K-bundle G for constructing Clifford module bundles over G/K, instead of (the double covering of) the frame bundle, (iii) the quantum Weil algebra and the cubic Dirac operator that comes with it, rather than the Riemannian Dirac operator. This results in a substantial simpliﬁcation in the proof of the local index theorem for compact homogeneous spaces.

8.1 REVIEWOFTHEHEATKERNELPROOFOF THELOCALINDEXTHEOREM

8.1.1 We quickly recapitulate the idea behind the heat kernel proof of the local index theorem. Let D/ be a Dirac operator on a vector bundle E over a compact, even-dimensional, spin manifold M. The space Γ(E) of smooth sections of E is naturally Z/2Z-graded (see Section 6.2.5): Γ(E) = Γ(E)+ ⊕ Γ(E)−. The Dirac operator is an odd operator: 0 D/ D/ = − . (8.1.2) D/ + 0 Assuming that D/ is symmetric1 (that is, it is equal to its formal adjoint), it enjoys the following analytic properties. (For proof, see [85, Ch.5].) It is essentially self-adjoint in the L2-closure2 Γ 2(E) of Γ(E), so we may assume that it is self-adjoint. Its spectrum form a discrete subset of iR. Each eigenvalue occurs with ﬁnite multiplicity, and the eigenfunctions are smooth. The space Γ 2(E) of square- integrable sections of E admits a Hilbert space direct sum decomposition into the eigenspaces of D/ . As a consequence, the image of D/ is equal to the orthogonal complement of the kernel of D/ . So the usual deﬁnition of the index as the difference between the kernel and the cokernel of D/ is trivial and uninteresting. So, when we speak of the index of a graded Dirac operator D/ , we mean the following graded index:

inds D/ := dim ker(D/ +) − dim ker(D/ −). ⊥ ± Now, since (ker D/ ) = im D/ , we have Γ(E) = ker(D/ ±) ⊕ im(D/ ∓). Thus, the graded index is equal to the usual index for the operator D/ +,

ind D/ + = dim ker(D/ +) − dim coker(D/ +).

1 This is the case for geometric Dirac operators (deﬁned by Equation 6.2.7); see [85, Prop. 3.11, p. 45]. As Equation 6.3.15 shows, the Kostant-Dirac operator is the 1 p geometric Dirac operator associated to the connection ∇X = ∂X + 3 γ (Xi). 2 Pick a Hermitian inner ( , ) product for E M that is invariant under the Clifford action of the tangent vectors (for existence, see Section 6.2.3); and deﬁne the L2- 1/2 inner product on Γ(E) as hσ1, σ2i = M(σ→1(x), σ2(x)) volx . R 119 8.1.3 MCKEAN-SINGER FORMULA. The kernel of D/ + and D/ − constitute the kernel of D/ :

ker D/ = ker D/ + ⊕ ker D/ −. The graded index of D/ is the difference between the dimensions of the subspaces in the above decomposition. Now observe that 2 ker D/ = ker D/ . (8.1.4) This follows from the self-adjointness of D/ , which implies hD/ 2σ, σi = hD/ σ, D/ σi. Thus, we have 2 2 inds D/ = dim ker(D/ )+ − dim ker(D/ )−,

2 2 ± where (D/ )± denote the restriction of D/ to the subspaces Γ(E) , 2 ± / : respectively. Denote the λ-eigenspace of D as Γ(E)λ and let Γ(E)λ = ± Γ(E)λ ∩ Γ(E) ; then + − inds D/ = dim Γ(E)0 − dim Γ(E)0 . Now we claim that, at least formally, we may add to the right-hand side the differences between the dimensions of nonzero eigenspaces:

+ − inds D/ = dim Γ(E)0 − dim Γ(E)0 " # + − + dim Γ(E)λ − dim Γ(E)λ . (8.1.5) 2 2 λ∈SpX(D/ )+ λ∈SpX(D/ )− λ6=0 λ6=0 The rationale behind this is that there is a complete super-symmetry between the even and the odd nonzero eigenmodes; in other words, 2 each nonzero eigenvalue of D/ occurs with same multiplicity in the even and the odd subspaces. This is because D/ maps an even λ- eigenfunction (λ 6= 0) to an odd one, and vice versa; indeed, if σ is ± / ∓ in Γ(E)λ , then D, as an odd operator, maps σ into Γ(E)λ , and D/ 2(D/ σ) = D/ (D/ 2σ) = λ(D/ σ). Now the formal equation 8.1.5 can be put into a more reasonable 2 form if we use the operator etD/ , t > 0. This begins with the obser- vation that 2 t(D/ )± ± tλ ± tr e = dim Γ(E)0 + e dim Γ(E)λ . 2 λ∈SpX(D/ )± λ6=0 The inﬁnite sums are well-behaved since the multiplicity of the eign- 2 values of D/ is at most of polynomial growth, which is a version of Weyl’s law for closed Riemannian manifolds [33]. It is also possi- 2 ble to directly verify that etD/ is of trace-class; see [85, Thm. 8.12,

120 p. 114]. The point is that we can replace the heuristic equation 8.1.5 with the following well-established equation:

2 2 2 t(D/ )+ t(D/ )− tD/ inds D/ = tr e − tr e = Str e . (8.1.6) This is known as the McKean-Singer formula. Here Str denotes the super-trace; it is deﬁned by Str(P) = tr(εP) where P is any trace-class operator and ε is the grading operator on Γ(E) which takes Γ(E)± as its ±1-eigenspace.

8.1.7 Here is another argument (from [12, p. 125]) for the McKean- 2 Singer formula. We begin by expressing Str etD/ in terms of the heat 2 kernel Qt of D/ . According to a general formula for operators with smooth integral kernels (see [85, Thm. 8.12, p. 114]), we have

tD/ 2 Str e = Str(Qt(x, x)) volx. (8.1.8) ZM Here Str(Qt(x, x)) is the super-trace of Qt(x, x) as an operator on Ex, and volx is the Riemannian volume form at x in M. Then,

d 2 2 Str etD/ = Str(∂ Q (x, x)) vol = Str(D/ Q (x, x)) vol , dt t t x t x ZM ZM where we have used the fact that the heat kernel satisﬁes the heat 2 equation. Note that D/ Qt(x, x) is the heat kernel of the operator 2 2 D/ etD/ . Thus, d 2 2 2 Str etD/ = Str(D/ etD/ ). (8.1.9) dt 2 Finally, keeping in mind that D/ commutes with etD/ , we have

2 1 2 2 1 2 2 D/ 2etD/ = D/ 2etD/ + D/ 2etD/ = D/ 2etD/ + D/ etD/ D/ 2 2 (8.1.10) 1 2 = [D/ , D/ etD/ ] , 2 s where [ , ]s is the super-commutator. Since the super-trace vanishes over super-commutators, Equations 8.1.9 and 8.1.10 imply d 2 Str etD/ = 0. dt 2 2 Thus, Str etD/ is constant for t > 0. As t 0, the operator etD/ 2 converges to the orthogonal projection P0 onto the kernel of D/ ; hence, we conclude that → tD/ 2 Str e = Str P0 = inds D/ . (8.1.11) The trick of Equation 8.1.10 can be used to prove a more general result; the following proposition is from [85, Prop. 11.9, p. 144]:

8.1.12 PROPOSITION. Let φ be any rapidly decreasing smooth function

121 on R with φ(0) = 1. Then, 2 Str φ(D/ ) = inds D/ .

2 Remark. The operator φ(D/ ) is deﬁned as the operator that acts as 2 the scalar φ(λ) on the λ-eigenspace of D/ in Γ 2(E).

2 Proof. Let P0 be the orthogonal projection onto the kernel of D/ . 2 2 Since the spectrum of D/ is discrete, we may write P0 = ψ(D/ ) for some smooth bump function ψ centered at 0 (so that ψ(0) = 1 and the support of ψ is compact). Then 2 inds D/ = Str P0 = Str ψ(D/ ).

Now let ϑ := φ − ψ; then 2 2 2 Str φ(D/ ) = Str ϑ(D/ ) + Str ψ(D/ ). 2 So the proposition follows once we show that Str ϑ(D/ ) = 0 holds for any rapidly decreasing function ϑ with ϑ(1) = 0. We may assume that ϑ(x) = xϑ¯(x) for some rapidly decreasing function ϑ¯. Then, ϑ(D/ 2) = D/ 2ϑ¯(D/ 2). 2 Since D/ commutes with ϑ¯(D/ ), we may rewrite the above as 1 ϑ(D/ 2) = D/ 2ϑ¯(D/ 2) + D/ ϑ¯(D/ 2)D/ . 2 2 Because D/ ϑ¯(D/ ) is an odd operator, we have 1 ϑ(D/ 2) = [D/ , D/ ϑ¯(D/ 2)] . 2 s 2 Therefore, Str ϑ(D/ ) = 0 as desired.

8.1.13 LOCAL INDEX THEOREM. By Equations 8.1.8 and 8.1.11, we have

inds(D) = Str(Qt(x, x)) volx. (8.1.14) ZM Because of this, Str(Qt) vol is called the index density of D/ . Since the left-hand side of this equation is independent of t, so must be the integral on the right-hand side. Examining, under this light, the small-t behavior of the index density is the heat kernel approach for the Atiyah-Singer index theorem. Let

i Qt ∼ Ht Ait ∞ i=0 X be the asymptotic expansion for Qt as t 0+. Here Ht is the Gaus- ∗ sian kernel, and the coefﬁcients Ai (i > 0) are in Γ(E E ). Since →

122 n/2 Ht(x, x) = 1/(2πt) , we have

1 i inds(D) ∼ t Str(Ai(x, x)) volx. (8.1.15) (2πt)n/2 ∞ i=0 M X Z The index theorem is achieved by showing that Str(Ai(x, x)) = 0 for i < n/2 and that top Str(An/2(x, x)) volx = Aˆ (M) ∧ ch(F) , x where the right-hand side denotes the top degree part of the exterior product of the the Hirzebruch Aˆ -class of M and the Chern character : of the auxiliary bundle F = EndCl(E). Put in another way, the index density satisfies the asymptotic equation top Str(Qt(x, x)) volx = Aˆ (M) ∧ ch(F) + O(t), x for t 0+. This statement alone (which implies the Atiyah-Singer index theorem) is known as the local index theorem. That→ a statement like the local index theorem might be true was suggested by H. P. McKean, Jr. and I. M. Singer [74, p. 61]. The first result in this direction is the work of V. K. Patodi [78], where he proved the Riemann-Roch-Hirzebruch theorem (which is a special case of the Atiyah-Singer index theorem) using the heat kernel method. V. K. Patodi’s work was computational in nature, and it was not clear why the vanishing of the lower order coefficients in the asymptotic expansion 8.1.15 occurs. Then, P. B. Gilkey [48] showed that the cancellations of high derivatives in the computation can be explained on the grounds of Invariant Theory. Developing on P. B. Gilkey’s idea, M. Atiyah, R. Bott, and V. K. Patodi [8] gave a proof for the local index theorem in its full generality; but Invari- ant Theory alone could not completely determine the coefficients, so they had to rely on computations of some examples. Finally, it was E. Getzler’s paper [47] that demonstrated the role of Clifford algebra in the vanishing of the lower order coefficients, and gave a completely analytic proof of the Atiyah-Singer index theorem. There are other proofs of the local index theorem that appeared later on, one of which is that of N. Berline and M. Vergne [13] which we mentioned in the beginning of this chapter.

8.1.16 Note that Ai(x, x) in the asymptotic expansion 8.1.15 takes value in End(Ex). In the case where n := dim M is even, we have a linear isomorphism Ex ' S ⊗ V where S is the spinor space for Cl(n) and V is some auxiliary vector space (see Section 5.1.5). The Z/2Z-grading of Ex arises from that of S. An element of End(Ex) can be identiﬁed with an element w ⊗ F P

123 in Cl(n) ⊗ End(V). And

Str(w ⊗ F) = trs(w) trV (F) where trs(w) is the super-trace of the action of w on S and trV (F) is the usual trace of F over V. We would like to mention at this point that the super-trace trs(w) is nonzero only if w contains a term that has top filtration order in Cl(n). To be more precise, let e1, . . . , en be an orthonormal basis for n R ; then, (−i)n/2, if I = {1, 2, . . . , n}, trs(eI) = (8.1.17) 0, otherwise, where eI are the elements in Cl(n) defined by Equation 5.1.9. To see why this is true, note first that the super-trace vanishes on super- commutators. If I 6= {1, . . . , n}, then eI is a super-commutator: eI = −[eIei, ei]s for any i not in I. To calculate trs(eI) for I = {1, . . . , n}, we use the fact that the space of even and odd spinors are the ±1- n/2 eigenspaces of the action of ω := (2i) e1 ··· en (see Section 5.1.44). Hence, n/2 2 trs(e1 ··· en) = tr(e1 ··· enω) = (2i) tr((e1 ··· en) ).

Since eiej = −ejei for i 6= j, 2 n(n−1)/2 2 2 (e1 ··· en) = (−1) e1 ··· en.

n(n−1)/2 n/2 2 Because (−1) = (−1) and ei = 1/2, we have

2 n/2 −n (e1 ··· en) = (−1) 2 . Thus, n/2 −n/2 trs(e1 ··· en) = (−i) 2 dim(S). n/2 Since dim(S) = 2 (see Section 5.1.5), n/2 trs(e1 ··· en) = (−i) .

8.2 THELOCALINDEXTHEOREMON G/K

8.2.1 NOTATIONS. We bring in the notations that we set up in Sec- tion 7.2.1. Thus, G is a compact connected Lie group with a bi- invariant metric, and K is a closed connected subgroup of G. We have a Lie group homomorphism Adf : K Spin(p) where p is the orthogonal complement of k in g. The induced Lie algebra homomorphism is → γp : k spin(p).

→

124 A formula for γp is given by Equation 5.4.10. We have a twisted Cl(p)-module E = S ⊗ V where S is the space of spinors for Cl(p) and the auxiliary space V is an irreducible K-vector space with highest weight µ. We denote the representation of K on V by π : K Aut(V). Thus, E is a K-vector space via → ν := Adf ⊗ π. The induced Lie algebra representation of ν is the differential of ν at the identity: p ν∗ = Adf ∗ ⊗ 1 + 1 ⊗ π∗ = γ + π∗. (8.2.2) p Each Lie algebra representation ν∗, γ , and π∗ extends to U(k) as an algebra homomorphism. Let us add few more notations that we shall use. For the rest of dim k dim p this chapter, { Xi }i=1 and { Yi }i=1 denote any orthonormal basis for k and p, respectively. For each vector X in g, we denote by X∗ the linear functional on g deﬁned by X∗(Y) = hX, Yi

∗ dim k ∗ dim p ∗ for Y in g. Hence, { Xi }i=1 and { Yi }i=1 serves as basis for k and p∗, respectively. We shall denote by λp the composition

γp q−1 k − spin(p) −− ∧2p where q−1 is the inverse of the Chevalley map 5.1.17. By Equa- tion 5.4.10, we have → → dim p 1 λp(X) = − hX, [Y ,Y ] iY Y . (8.2.3) 2 i j g i j i,j=1 X We shall often use the notation

x¯ := xK for the points in G/K. Finally, recall that our model for the tangent bundle T(G/K) is T ×K p (see Section 6.2.8). This means that we are identifying the ∗ tangent space Te¯(G/K) and the cotangent space Te¯ (G/K) with p and p∗, respectively.

8.2.4 As far as index theory is concerned, R. Bott showed that a nontrivial index can occur only if K is of maximal rank (that is, K contains a maximal torus of G)[16, Thm. II, p. 170]. So we shall concentrate on the case where K is of maximal rank. Two implica-

125 tions of this are:

(i) The dimension of G/K, which is equal to that of p, is even. To see this, pick a common maximal torus for G and K so that their Lie algebras share the same maximal commutative subalgebra in their root space decomposition. Let Φg and Φk be the set of roots of G and K, respectively. Then Φk is a subset of Φg, and the dimension of p is equal to the cardinality of Φg \ Φk, which is even, since the roots come in positive and negative pairs (see Section 2.3.11).

(ii) Because dim(p) is even, the spinor space S for Cl(p) is naturally Z/2Z-graded (see Section 5.1.44). This gives rise to Z/2Z- gradings on the Clifford module E, the associated vector bundle G ×ν E, and the space of smooth sections of G ×ν E.

Example (The Flag Manifold G/T). As a special case, suppose K is equal to a maximal torus T of G. Then G/T is spin [41, Cor.1.12, p. 91]. The homomorphism Adf : T Spin(p) (8.2.5) gives rise to a T-action on the spinor space S. The complexification + of p is the union of the root spaces→ of g. Let Φ = {α1, . . . , αk} be the selected set of positive roots of g. (Hence, k = dim p/2.) As we shall soon see, the set of the weights of the T-representation on S is 1 k (β + ··· + β ):(β , . . . , β ) ∈ {±α } , (8.2.6) 2 1 k 1 k i i=1 Y k : 1 the highest weight being ρ = 2 i=1 αi; for the multiplicities, the k-tuple (β1, . . . , βk) contributes with multiplicity 1 to the weight 1 P k 2 (β1 + ··· + βk). So dim(S) = 2 , which is consistent with the general theory of spinors. The weights and their multiplicities can be verified as follows. (The line of argument is from [68, p. 11].) The complexification of p is simply the collection of all the root spaces: M pC = gC,α ⊕ gC,−α . α∈Φ+

Choose vectors Yα in gC,α for each positive root α, and let Y−α be the vector such that hYα,Y−αi = 1; since the root spaces gC,α and gC,β are orthogonal if α 6= ±β, the vector Y−α is necessarily in gC,−α. Their collection { Y±α }α∈Φ+ form a basis for pC. To ﬁnd the weights of the T-representation on S, we need to examine the induced Lie algebra representation, which is given by γp : t spin(p). By Equa- tion 5.4.10,

p 1 → γ (H) = adH(Yα)Y−α + adH(Y−α)Yα . 2 + αX∈Φ

126 Since Y±α are root vectors, satisfying adH(Y±α) = ±α(H)Y±α,

p 1 γ (H) = iα(H)(YαY−α − Y−αYα). 2 + αX∈Φ Since YαY−α = −Y−αYα + hYα,Y−αi in Cl(p) and hYα,Y−αi = 1,

p 1 γ (H) = iα(H) − iα(H)Y−αYα 2 + + αX∈Φ αX∈Φ = iρ(H) − iα(H)Y−αYα. (8.2.7) + αX∈Φ The Clifford relation can be used in the other way around to get

p γ (H) = −iρ(H) + iα(H)YαY−α. (8.2.8) + αX∈Φ Now, if v is a weight vector in S with weight λ, then the action of Y±α maps v to a weight vector with weight λ ± α; this owes to the following super-commutator relation: p [γ (H),Y±α] = adH(Y±α) = ±iα(H)Yα. (The ﬁrst equality is just Equation 5.1.54.) Now suppose v is a weight vector with highest weight. Then Yα · v must be zero, and hence, by Equation 8.2.7, γp(H) · v = iρ(H)v. Therefore, ρ is the weight of v, which is the highest weight of the T-action on S. Similar argument using Equation 8.2.8 tells us that the lowest weight is −ρ, which is equal to ρ − α∈Φ+ α. Hence, + ρ − α∈I α for I ⊆ Φ gives us all the weights, each I contributing with multiplicity 1. P P 8.2.9 Let ∼ K Ξ : Γ(G ×ν E) − (C (G) ⊗ E) be the vector space isomorphism deﬁned∞ in Section 6.3.1. By Proposi- → tion 6.3.14, Ξ intertwines the differential operators D/ g/k on Γ(G×µ E) and D/ (g, k) on (C (G) ⊗ E)K. Now, by Theorem 7.2.4, D/ (g, k)2 is equal to ∞ 1 1 2 2 L := Ωb g + kµ + ρkk − kρgk 2 2 / 2 as operators on (C (G) ⊗ E)K. Hence, Ξ intertwines etDg/k and etL.

∞ tL (C (G) ⊗ E)K e / (C (G) ⊗ E)K O O ∼ ∼ ∞ Ξ ∞ Ξ (8.2.10)

Γ(G ×ν E) / Γ(G ×ν E) tD/ e g/k

127 2 We wish to compare the heat kernel of D/ g/k with that of L. To that K end, recall that a 2-tuple (σ,¯ σ) in Γ(G ×ν E) × (C (G) ⊗ E) is in the graph of Ξ if and only if ∞ σ¯(g¯) = [g, σ(g)] (8.2.11) for any g in G. With this in mind, we adopt the following deﬁnition:

8.2.12 DEFINITION. Let Qt (t > 0) be an integral kernel for some operator on Γ(G ×ν E), where G/K is endowed with the quotient measure (see Section 2.3.24). Consider a function

Q˜ t : G × G End(E) that satisﬁes the following two conditions: → (i) For all (k1, k2) in K × K, −1 Q˜ t(xk1, yk2) = ν(k1) Q˜ t(x, y)ν(k2). (8.2.13)

(ii) For any (σ,¯ σ) in the graph of Ξ, h i Qt(x,¯ y¯)σ¯(y¯) dy¯ = x, Q˜ t(x, y)σ(y) dy¯ . (8.2.14) ZG/K ZG/K (In the above, [g, v] is the notation for the K-orbit of (g, v) in G × E.)

Condition (i) is necessary for Equation 8.2.14 to make sense. If such Q˜ t exists, we shall call it the kernel on G that is equivalent to Qt.

1 1 2 2 8.2.15 LEMMA. Let Rt be the heat kernel of 2 Ωb g + 2 (kµ+ρkk −kρgk ) on (C (G) ⊗ E)K. −1 ∞ Rt(xk1, yk2) = ν(k1) Rt(x, y)ν(k2) for (x, y) in G × G and (k1, k2) in K × K.

Remark. A version of this can be found in [13, Eq. 3.18, p. 333].

Proof. Let σ be an arbitrary element in (C (G) ⊗ E)K. Then σ(yk) = ν(k)−1σ(∞y) for y in G and k in K. Thus,

−1 Rt(x, yk)ν(k) σ(y) dy = Rt(x, yk)σ(yk) dy ZG ZG −1 = Rt(x, y)σ(y) d(yk ) = Rt(x, y)σ(y) dy. ZG ZG This proves that −1 Rt(x, yk)ν(k) = Rt(x, y). Next, since the map

x 7 Rt(x, y)σ(y) dy ZG → 128 deﬁnes an element of (C (G) ⊗ E)K, we have

∞ −1 Rt(xk, y)σ(y) dy = ν(k) Rt(x, y)σ(y) dy. ZG ZG This proves that −1 Rt(xk, y) = ν(k) Rt(x, y).

2 8.2.16 LEMMA. Let Qt be the heat kernel of D/ g/k on Γ(G ×ν E). Let Rt 1 1 2 2 K be the heat kernel of 2 Ωb g + 2 (kµ + ρkk − kρgk ) on (C (G) ⊗ E) . Then, ∞ −1 Q˜ t(x, y) := Rt(x, yk)ν(k) dk (8.2.17) ZK deﬁnes an integral kernel on G that is equivalent to Qt.

Remark. A version of this can be found in [13, Prop. 3.20, p. 333].

Proof. Let k1 and k2 be arbitrary elements of K. By Lemma 8.2.15 and the invariance of the measure on K, we have

−1 Q˜ t(xk1, yk2) = Rt(xk1, yk2k)ν(k) dk ZK −1 −1 −1 −1 = ν(k1) Rt(x, yk)ν(k2 k) d(k2 k) ZK −1 −1 = ν(k1) Rt(x, yk)ν(k) ν(k2) dk K Z −1 = ν(k1) Qt(x, y)ν(k2). This checks property 8.2.13. It remains to check Equation 8.2.14. Let (σ,¯ σ) be an arbitrary 2-tuple in the graph of Ξ. Let x be an arbitrary point in G. Suppose

/ 2 (etDg/k σ¯)(x¯) = [x, w]. (8.2.18) We need to show that

w = Q˜ t(x, y)σ(y) dy.¯ (8.2.19) ZG/K / 2 2 2 Since (etDg/k σ,¯ et(Ωb g+kµ+ρkk −kρgk )/2σ) is in the graph of Ξ, Equa- tions 8.2.11 and 8.2.18 imply

2 2 w = (et(Ωb g+kµ+ρkk −kρgk )/2σ)(x). This can be rewritten in terms of the heat kernel as

w = Rt(x, y)σ(y) dy. ZG Then, by the fact that

f(g) dg = f(gk) dk dg¯ ZG ZG/K ZK for any continuous function f on G (see [34, Eq. 3.13.19, p.184]),

129 we have

w = Rt(x, yk)σ(yk) dk dy.¯ ZG/K ZK Since σ(yk) = ν(k)−1σ(y),

−1 w = Rt(x, yk)ν(k) σ(y) dk dy¯ ZG/K ZK

= Q˜ t(x, y)σ(y) dy.¯ ZG/K This is Equation 8.2.19, which proves that Q˜ t satisﬁes Equation 8.2.14.

8.2.20 COROLLARY. Let Q˜ t and Rt be as in Lemma 8.2.16. Let q˜t and rt be the convolution kernels associated to Q˜ t and Rt, respectively. Then

−1 q˜t(g) = rt(gk)ν(k) dk ZK for any g in G.

Proof. By deﬁnition (see Section 7.1),

q˜t(g) = Q˜ t(e, g),

rt(g) = Rt(e, g). So, by Lemma 8.2.16,

−1 −1 q˜t(g) = Rt(e, gk)ν(k) dk = rt(gk)ν(k) dk. ZK ZK 8.2.21 NOTATION. Recall that the orthogonal decomposition g = k⊕p k is an ad(k)-invariant decomposition. For each X in k, let adX and p adX denote the restrictions of adX to k and p, respectively. Then k adX 0 the matrix for adX is a block diagonal matrix of the form p . 0 adX Hence the power series sinh(ad /2) j(X) = det1/2 X adX /2 factors into j(X) = jk(X)jg/k(X) (8.2.22) where jk and jg/k are deﬁned by k p 1/2sinh(adX /2) 1/2sinh(adX /2) jk(X) := det p , jg/k(X) := det p . adX /2 adX /2 (8.2.23)

8.2.24 LEMMA. Let q˜t be as in Corollary 8.2.20. For t 0+, we have an asymptotic equality

2 jk(X) → q˜ (e) ∼ etkρk+µk /2 hg(X)e−ν∗(X) dX t j (X) t Zk g/k

130 as End(E)-valued functions of t.

Proof. By Theorem 7.2.15 and Corollary 8.2.20, we have 2 2 jk(X) q˜ (e) ∼ etkρk+µk /2 hg(X)e−ν∗(X) dX. t j(X) t Zk The assertion then follows from Equation 8.2.22.

The following lemma is from [35, Lem. 11.3, p. 137]:

8.2.25 LEMMA. Let ϕ be a smooth function on k with sufﬁciently slow growth. Then, the ∧(p)-valued function

k −λp(X) t 7 ht(X)ϕ(X)e dX Zk n has an asymptotic expansion→ n=0 t Ψn for t 0+. The nth co- Ln 2` efﬁcient Ψn is contained in `=0∞∧ (p). If n 6 dim(p)/2, then the 2n P component of Ψn in ∧ (p) (the highest degree part)→ is given by dim k n (2n) 1 p Ψn = −λ (Xi)∂i ϕ(X) , (8.2.26) n! X=0 i=1 X where ∂i is the partial derivative with respect to the ith coordinate variable xi : X 7 hXi,Xi.

Proof. The Gaussian function hk is the heat convolution kernel of the → t generalized Laplacian ∆ˆk/2. Thus, for any smooth function ψ on k with sufﬁciently slow growth,

t∆ˆk/2 k (e ψ)(0) = ht(X)ψ(X) dX. (8.2.27) Zk tn 1 ˆn This is asymptotically equal to n=0 n! ( 2n ∆k ψ)(0) as t 0+. (This is fairly easy to check by substituting ψ with its Taylor series on P∞ the right-hand side of Equation 8.2.27 and applying the formula 2 √ → e−x /2x2k dx = 2π(2k)!/(k!); see [12, Prop. 2.13, p. 73] for de- R tails.) Hence, R k −λ(X) n ht(X)ϕ(X)e dX ∼ t Ψn, ∞ k n=0 Z X where ˆ n 1 ∆k −λ(X) Ψn := (ϕ(X)e ) . n! 2 X=0

131 Now ∆ˆ 1 dim k k (ϕ(X)e−λ(X)) = Xˆ Xˆ (ϕ(X)e−λ(X)) 2 2 i i i=1 X 1 dim k 1 = ∂2ϕ(X)e−λ(X) + 2∂ ϕ(X)∂ e−λ(X) + ϕ(X)∂2e−λ(X) 2 2 i i i i i=1 X 1 dim k = ∂2ϕ(X) − 2∂ ϕ(X)λ(X ) + 4ϕ(X)λ(X )λ(X ). 2 i i i i i i=1 X The last term is zero because λ(Xi)λ(Xi) = 0. Thus,

∆ˆ hdim k 1 i k (ϕ(X)e−λ(X)) = ∂2 − λp(X )∂ ϕ(X) e−λ(X). 2 2 i i i i=1 X Therefore, dim k ∆ˆ n h 1 n i k (ϕ(X)e−λ(X)) = ∂2 − λp(X )∂ ϕ(X) e−λ(X), 2 2 i i i i=1 X and hence, dim k n 1 1 2 p Ψn = ∂i − λ (Xi)∂i ϕ(X) . n! 2 X=0 i=1 X 2 Since λ(Xi) is in ∧ (p), we see that the exterior algebra factor of the terms in Ψn are all of even degree at most 2n. And, if 2n 6 dim(p), the component of Ψn that has degree 2n is given by Equation 8.2.26.

8.2.28 Let ϕ be as in Lemma 8.2.25. The nth order term of the Taylor series for ϕ(X) with respect to X = 0 is

dim k 1 n xi∂i ϕ(X) . n! X=0 i=1 X As observed by N. Berline and M. Vergne [13, § 1.12, p. 314], substi- p tuting xi with −λ (Xi) in the above expression gives the right-hand side of Equation 8.2.26. Now the coordinate variable xi is just the ∗ linear functional Xi . Hence, the Taylor series yields an algebra ho- ∗ momorphism C (k) R[[k ]], where the codomain is the algebra of ∗ formal power series∞ in k over R; composing this with the algebra ∗ ∗ p homomorphism R[[k →]] ∧(p) generated by X 7 ∧ (X) yields an algebra homomorphism → C (k) ∧(p). → (8.2.29) Note that this homomorphism∞ factors through the symmetric algebra S(k∗). →

8.2.30 NOTATION. Let f be a smooth function on k. We shall denote

132 its image under the homomorphism 8.2.29 by f(−λpX).

8.2.31 DEFINITION. Let p be the projection map of the principal K- bundle G G/K. Let U be a locally trivializing neighborhood containing the identity cosete ¯ in G/K; let ξ : p−1(U) U × K be the trivialization→ over U. Consider the composition map ξ → p−1(U) − U × K K, where the second map is the projection onto the second component. We shall denote by θ the pullback→ of→ the Maurer-Cartan connection on K along the above composition, and call it the local Maurer- Cartan connection near the identity in the principal K-bundle G (see the example on page 94). At the identity, θ is the k-valued 1-form dim k ∗ ∗ θ = Xi ⊗ Xi ∈ k ⊗ g . i=1 X The curvature of θ shall be denoted by Θ. At the identity, Θ is of the form dim k i 2 ∗ Θ = Xi ⊗ Θ ∈ k ⊗ ∧ (g ), i=1 X where Θi is a horizontal 2-form. Note that the horizontal subspace of g with respect to θ is p. Hence, each Θi is contained in the subalgebra ∧2(p∗) of ∧2(g∗).

8.2.32 NOTATION. The local Maurer-Cartan connection θ on the principal K-bundle G G/K induces local connections for the tangent bundle T ×K p and the twisting bundle G ×K V over G/K (see Sec- tion 6.1.29). We shall→ denote their curvature 2-forms, respectively, by ΘT and ΘV ; at the identity cosete ¯, they take the following form: dim k p i 2 ∗ ΘT = − ad (Xi) ⊗ Θ ∈ End(p) ⊗ ∧ (p ), (8.2.33) i=1 X dim k i 2 ∗ ΘV = − π∗(Xi) ⊗ Θ ∈ End(V) ⊗ ∧ (p ). (8.2.34) i=1 X 8.2.35 LEMMA. Let Θi be deﬁned as in Deﬁnition 8.2.31. Under the algebra isomorphism ∧(g) ∧(g∗), X 7 X∗, we have p i λ (Xi) 7 Θ . → → Proof. Since the 2-form Θi is in ∧2(p∗), we may write it as → dim p 1 Θi = Θi(Y ,Y )Y∗Y∗. 2 a b a b a,b=1 X

133 Equation 6.1.15 tells us that i ∗ Θ (Ya,Yb) = −Xi ([Ya,Yb]g) = −hXi, [Ya,Yb]gi. Therefore, dim p 1 Θi = − hX , [Y ,Y ] iY∗Y∗. 2 i a b g a b a,b=1 X Comparing this with Equation 8.2.3, we see that the above is exactly p ∗ the image of λ (Xi) under the isomorphism ∧(g) ∧(g ), X 7 X∗. → → 8.2.36 DEFINITION. We define the algebra homomorphism ∗ A : C (k) ⊗ C ∧(p ) ⊗ C as the C-linear extension∞ of the composition → C (k) ∧(p) ∧(p∗), where the first map is given∞ by f 7 f(−λpX), and the second map is the algebra isomorphism induced→ by →Y 7 Y∗. → 8.2.37 PROPOSITION. Let ϕ be the element in C (k) ⊗ defined by → C the following power series: ∞

: −1 −π∗X ϕ(X) = jk(X)jg/k(X) trV (e ).

Here trV denotes the usual trace for the operators on the representation space V of π. Then, Θ /2 1/2 T −ΘV A(ϕ) = det trV (e ). (8.2.38) sinh ΘT /2 Proof. By deﬁnition (Equation 8.2.23), p 1/2sinh(adX /2) jg/k(X) = det p . adX /2 Then, by Lemma 8.2.35 and Equation 8.2.33,

−1 1/2 ΘT /2 A(jg/k) = det . (8.2.39) sinh ΘT /2 By the same token,

−π∗ −ΘV A trV (e ) = trV (e ).

It remains to show that A(jk) = 1. Because jk(X) is invariant under the Ad(K)-action on k, its value is completely determined by jk(H) for H in a maximal abelian subalgebra of k; such a subalgebra is the Lie algebra t of a maximal torus of K. Hence, it is sufﬁcient to show that p + jk(−λ H) = 1 for H in t. Let Φ denote the selected set of positive roots of k. Then sinh α(H/2) jk(H) = . + α(H/2) αY∈Φ

134 dim t dim t Let { Hi }i=1 be an orthonormal basis for t. Writing H = i=1 xiHi, we have sinh( dim t x α(H )/2) P j (H) = i=1 i i . k dim t + i=1 xiα(Hi)/2 αY∈Φ P Thus, P dim t p p sinh( i=1 λ (Hi)α(Hi)/2) jk(−λ H) = dim t p + i=1 λ (Hi)α(Hi)/2 αY∈Φ P p dim t p sinhP(λ ( α(Hi)Hi)/2) sinh(λ (H˜ )) = i=1 = , p dim t p ˜ + λ ( i=1 α(Hi)Hi)/2 + λ (H) αY∈Φ P αY∈Φ dim t ˜ : 1 P where H = 2 i=1 α(Hi)Hi. Since the power series of sinh(x)/x involves only the even powers of x and λp(H˜ )λp(H˜ ) = 0, we have p P jk(−λ H) = 1.

8.2.40 THEOREM(THELOCALINDEXTHEOREMFOR G/K). Let G be a compact connected Lie group equipped with a bi-invariant metric. Let K be a closed connected Lie subgroup of G such that G/K is an even-dimensional spin manifold. Let g and k be the Lie algebra of G and K, respectively, and let p be the orthogonal complement of k in g. Consider the Kostant-Dirac operator D/ g/k on the twisted spinor bundle G ×K (S ⊗ V) G/K, where S is the spinor space for Cl(p) and V is an irreducible representation space for K. Let qt be the heat convolution 2 kernel of D/ g/k.→ The leading nonzero term of the asymptotic expansion of the super-trace of qt, multiplied by the Riemannian volume form of G/K, is equal to the top degree part of the product of the Hirzebruch Aˆ -class of G/K and the Chern character of the twisting bundle G ×K V; that is, top Str(qt) vol = Aˆ (G/K) ch(G ×K V) + O(t), for t 0+.

Proof. By the homogeneity of G/K, it is sufﬁcient to check for Str(q ) → t ate ¯ := eK. Letq ˜t be the convolution kernel on G that is equivalent to qt (see Corollary 8.2.20); then,

Str(qt(e¯)) = Str(q˜t(e)). (8.2.41) By Lemma 8.2.24 and Equation 8.2.2, we have

2 p tkρk+µk /2 g −1 −π∗(X) −γ (X) q˜t(e) ∼ e ht (X)jk(X)jg/k(X)e e dX. Zk By definition, γp = q ◦ λp, where q is the Chevalley map ∧(p) Cl(p). Recall that q is an algebra homomorphism modulo terms of lower filtration order. As we have seen in Equation 8.1.17, the super-→ trace is dependent only on the elements with top filtration order. So, p as far as the super-trace is concerned, we may replace e−γ (X) in the

135 p integrand with e−λ (X). Thus,

Str(qt(e¯)) = Str(q˜t(e)) tkρ +µk2/2 e k p ∼ hk(X)j (X)j−1 (X) tr (e−π∗(X)) tr (e−λ (X)) dX, (2πt)dim p/2 t k g/k V s Zk where trs is the super-trace over S, and trV is the ordinary trace over V. Applying Lemma 8.2.25 with

: −1 −π∗X ϕ(X) = jk(X)jg/k(X) trV (e ) leads us to the following conclusions:

(i) We have the asymptotic expansion

2 tkρk+µk /2 e n Str(qt(e¯)) ∼ t trs(Ψn), (8.2.42) (2πt)dim(p)/2 ∞ n=0 X Ln 2` where Ψn is contained in `=0 ∧ (p). (ii) Since the super-trace of an element in ∧kp can be nonzero only if k is the top degree, which means k = dim p in our case, the leading nonzero term in the asymptotic expansion 8.2.42 comes from the term with n = dim(p)/2. Hence, 1 dim p Str(q (e¯)) = tr (Ψ ) + O(t),N := . (8.2.43) t (2π)N s N 2

(iii) Owing to Equation 8.2.26, p trs(ΨN) = trs(ϕ(−λ X)). (8.2.44)

Now the super-trace picks out the top degree part; precisely, by Equa- tion 8.1.17, N top trs(ΨN)Y1 ··· Y2N = (−i) ΨN . Hence, Equation 8.2.44 implies p N p top trs(ΨN)Y1 ··· Y2N = trs(ϕ(−λ X))Y1 ··· Y2N = (−i) ϕ(−λ X) . Applying the algebra isomorphism ∧(p) ∧(p∗), Y 7 Y∗, we get ∗ ∗ N top trs(ΨN)Y1 ··· Y2N = (−i) A(ϕ) . → → ∗ ∗ Since Y1 ··· Y2N is the Riemannian volume form ate ¯, we have 1 1 tr (Ψ ) vol = A(ϕ)top. (2π)N s N e¯ (2πi)N By Proposition 8.2.37, 1 1 h Θ /2 itop 1/2 T −ΘV N trs(ΨN) vole¯ = N det trV e (2π) (2πi) sinh ΘT /2 e¯ = Aˆ (G/K) (G V) top. ch ×K e¯ (8.2.45) The theorem now follows from Equation 8.2.43.

136 THEDISTRIBUTIONALINDEXOF9 THECUBICDIRACOPERATOR

HE relative Dirac operator D/ (g, k), identified as a differential op- T erator following the prescription in Section 6.3, is a priori, elliptic only in the directions transverse to the K-orbits in G. Hence, the kernel of D/ (g, k), which is a K-representation space, may not be finite-dimensional. It turns out, however, that for each irreducible representation u of K, the multiplicity nu of the representation space Vu of u in ker D/ (g, k) is finite [6, Thm. 2.2, p. 10]. So we can associate to ker D/ (g, k) an infinite formal sum n [V ] in the formal u∈Kb u u representation group R(K) = · [V ]. (We had a brief en- b u∈Kb Z Pu R(K) D/ (g, k) counter with b on page 37.)Q Assuming that is graded, we write + − [ker D/ (g, k)+] := nu [Vu], [ker D/ (g, k)−] := nu [Vu], uX∈Kˆ uX∈Kˆ + − where nu and nu are the multiplicities of Vu in the even and odd subspaces of ker D/ (g, k), respectively. Then, set

+ − IndK D/ (g, k) := [ker D/ (g, k)+] − [ker D/ (g, k)−] = (nu − nu )[Vu]. uX∈Kˆ According to the results of M. Atiyah and I. M. Singer [6, Thm. 2.2, p. 10, Thm. 4.6, p. 34], this sum converges, in the distributional sense, to a distribution on K that is supported at the identity. Hence, we may apply the inverse of the Duﬂo isomorphism to IndK D/ (g, k) −1 and obtain a distribution Duf IndK D/ (g, k) on k that is supported at 0. Our goal is to prove Theorem 9.2.17, which says that, for any ad(k)-invariant polynomial φ on k, −1 ˆ ˆ hDuf IndK D/ (g, k), φi = hA ` φ, [G/K]i,

137 where [G/K] is the fundamental homology class of G/K, and Aˆ ` φˆ is the cup product of the Hirzebruch Aˆ -class of G/K and the characteristic class φˆ of the principal K-bundle G G/K given by the image of φ under the Chern-Weil homomorphism S(k)k H∗(G/K). → 9.1 THEDISTRIBUTIONALINDEXOF → ATRANSVERSALLYELLIPTICOPERATOR

9.1.1 We review, in this section, the basics of transversally elliptic operators. Main reference is [6].

9.1.2 NOTATION. Throughout this chapter, K is a compact Lie group and M is a manifold with a K-action. The cotangent bundle of M shall be denoted by ξ : T ∗M M. And we detnote by D a generic K-equivariant differential operator on a K-equivariant1 vector bundle F→ M.

9.1.3 DEFINITION. We deﬁne the subset T ∗M of T ∗M as → K ∗ ∗ TKM := { α ∈ T M | α(Xeξ(α)) = 0, ∀X ∈ k }.

Here Xe denotes the fundamental vector ﬁeld on M generated by X (see Equation 6.1.4).

∗ Remark. The restriction of ξ to TKM makes it a ﬁber bundle over M; each ﬁber is a vector space, yet the dimension may vary from point to point on M.

9.1.4 DEFINITION. A K-equivariant differential operator D on a K- equivariant vector bundle F M is transversally elliptic if its principal symbol (which is a bundle morphism ξ∗F ξ∗F) is invertible on ∗ TKM outside (the image of)→ the zero section. → Remark. (1) The deﬁnition of the principal symbol using pseudodifferential operator theory can be found in [6, Lecture 1].

(2) Elliptic operators are always transversally elliptic. If K is ﬁnite, then transversal ellipticity is equivalent to ellipticity.

9.1.5 By the standard theory of pseudodifferential operators, a transversally elliptic operator D : Γ(F) Γ(F) extends to an operator on Sobolev spaces; see [90, Thm. 5.1, p. 47]. → 9.1.6 THEOREM. Let D be a K-equivariant differential operator on a K-equivariant vector bundle F M. If D is transversally elliptic, then,

1 A K-equivariant vector bundle is a vector bundle F M such that (i) the total space F and the base space M are K-manifolds,→ (ii) the projection F M is K-equivariant, and (iii) the K-action on the ﬁbers are linear. → →

138 for each u in Kb, the u-isotypic component of ker D and ker D∗ are ﬁnite-dimensional and consist of smooth sections. If we denote the ∗ + − multiplicities of u in ker D and ker D by nu and nu , respectively, and denote the character of u by χu, then the sum + − (nu − nu )χu uX∈Kb converges as a distribution on K.

A Comment on the Proof. This theorem is from [6, Thm. 2.2, p. 10]; the actual statement there is more general then what we wrote down above. We will not repeat the proof here, but introduce some ideas dim k and constructions that are involved. Let { Yi }i=1 be an orthonormal basis for k. Let Y˜i denote the fundamental vector field on M generated by Yi. Define the differential operator ∆K on M as dim k ∆K := Y˜iY˜i. i=1 X Consider the operator A : Γ(F) Γ(F) ⊕ Γ(F) σ 7 (Dσ, (1 − ∆K)σ). → Because the symbol of 1 − ∆K is injective in the direction of K-orbits, the symbol of the operator A→is injective. Then, ∗ ∗ 2 A A = D D + (1 − ∆K) is an elliptic operator on Γ(F). Define

(ker D)λ = { σ ∈ Γ(F) | Aσ = 0 ⊕ λσ }

= { σ ∈ Γ(F) | Dσ = 0, ∆Kσ = (1 − λ)σ }.

Then, for any σ in (ker D)λ, we have (A∗A)σ = λ2σ. 2 ∗ Thus, (ker D)λ is contained in the λ -eigenspace of A A. But the eigenspaces of A∗A are ﬁnite-dimensional and consists of smooth sections, owing to the ellipticity of A∗A [6, Lem. 2.3, p. 10]; hence, so must the subspaces (ker D)λ be. Now (ker D)λ contains all the u- isotypic components of ker D on which the Casimir element acts by the scalar λ (see Sections 2.2.14 and 2.2.24). Therefore, we conclude that the multiplicities of all irreducible representations of K in ker D must be ﬁnite. A similar argument works for ker D∗.

9.1.7 DEFINITION. Owing to Theorem 9.1.6, ker D and ker D∗ deﬁne the following elements in the formal representation group Rb(K) = · [V ]: u∈Kb Z u + ∗ − Q [ker D] = nu [Vu], [ker D ] = nu [Vu]. (9.1.8) uX∈Kb uX∈Kb

139 Their difference, ∗ IndK D := [ker D] − [ker D ], (9.1.9) is called the distributional index of D.

Remark. If D is elliptic, then [ker D] and [ker D∗] are ﬁnite sums, and they lie in the character ring R(K). So their difference IndK D is also an element of R(K). In this case IndK D is known as the equivariant index of D.

9.1.10 Another way to deﬁne the distributional index is to specify its pairing with the test functions f in C (K). Let R : K Aut(Γ(F)) be the K-action on the space of smooth∞ sections of F with respect to which the operator D is equivariant. Deﬁne the operator→Rf that acts on Γ(F) by

(Rfσ)(x) = f(k)(Rkσ)(x) dk. (9.1.11) ZK + − ∗ Let Rf and Rf be the restrictions of Rf to ker D and ker D , respectively. Then the pairing of IndK D with f in C (K) is given by

+ ∞− hIndK D, fi = tr(Rf ) − tr(Rf ). (9.1.12) See [6, p. 9–10] for more discussions on this.

Remark. Let ( , ) be a smooth ﬁber-wise inner product on the vector bundle F M. Let µ be any density on M. Deﬁne an L2-inner product on Γ(F) by → h i1/2 hσ1, σ2i = (σ1(x), σ(x)) µ(x) . ZM We may assume that this inner product is K-invariant by the usual trick of averaging the inner product over K (see Section 2.2.2). Then,

hRkσ, Rkσi = hσ, σi for any σ in Γ(F) and k in K; as a consequence, the linear map Rk extends, as a unitary operator, to the L2-closure Γ 2(F) of Γ(F). Then, for each f in C (K), the operator Rf is a bounded operator; indeed, for any σ in Γ(F∞) and x in M, we have

|Rfσ(x)| 6 |f(k)| kRkk kσk dk, ZK which implies

kRfk 6 |f(k)| dk. ZK The following theorem is from [6, Thm. 4.6, p. 34]:

9.1.13 THEOREM. The support of the distribution IndK D is contained in the subset of K comprising of all elements that admit at least one

140 ﬁxed point in M; put in another way, [ supp(IndK D) = { h ∈ K | x = x · h }. (9.1.14) x∈M

9.1.15 COROLLARY. If K acts freely on M then IndK D is supported at the identity of K.

Example. Let M be a compact connected Lie group G. Let K be a maximal torus T of G, and let it act on G on the left. Let p be the orthogonal complement of k in g with respect to some Ad(G)- invariant inner product. Let S be the spinor space for the Clifford algebra Cl(p). Take the trivial bundle G × S G, and let T act on the sections by the left-regular action. That means, for h in T and σ in Γ(G × S), → h · σ : g 7 σ(h−1g). Let D/ be the cubic Dirac operator 5.2.10; following exactly what we have done in Section 6.3.5, we identify→ D/ as a T-equivariant differential operator on Γ(G × S) = C (G) ⊗ S. As we have discussed in Section 8.2.4, G/T is even-dimensional∞ and the spinor space is bi- + − graded: S = S ⊕ S . This induces a grading on Γ(G × S), and the / Dirac operator is an odd operator: D/ = 0 D− . We are interested D/ + 0 in the distributional index IndT D/ + of the (transversally) elliptic op- ∗ erator D/ +. Since D/ − = D/ + (see Section 8.1.1), we have

IndT D/ + = [ker D/ +] − [ker D/ −] ∈ Rb(T).

Let Cµ denote the irreducible complex T-representation space with weight µ. Recall that the correspondence Cµ 7 µ is an isomorphism from Tb to the weight lattice ΛT (see Section 2.3.4). Decomposing the T-vector space Γ(G × S) into isotypic components,→ we get ± M ± Γ(G × S ) ' Cµ ⊗ Hom(Cµ,C (G) ⊗ S ) µ∈Λ T ∞ M ± ∗ T ' Cµ ⊗ (C (G) ⊗ S ⊗ Cµ) . µ∈Λ T ∞ Thus, [ker D/ ±] = dim ker(D/ µ±)[Cµ] µ∈Λ XT where D/ µ is the Kostant-Dirac operator on the space of T-invariant ∗ sections of the trivial bundle G × (S ⊗ Cµ) G. Hence, Ind D/ = ind D/ [ ], (9.1.16) T + s µ→Cµ µ∈Λ XT where inds D/ µ is the usual graded index of D/ µ. Let W be the Weyl group of G, and let ρ be half the sum of the

141 positive roots of G. For µ in ΛT , set

Alt(µ) := sgn(w) w · µ ∈ ΛT . w∈W X By the properties of the weight lattice summarized in Sections 2.3.11 and 2.3.15, Alt(µ) is nonzero only if W acts freely on µ, that is, µ is off the walls of the Weyl chambers in t∗; and if that is the case, there is a unique dominant weight λ in ΛT such that Alt(λ + ρ) is equal to Alt(µ) up to a sign. According to a result of R. Bott [16, § 6–7],

dim(Vλ), if Alt(µ) = −Alt(λ + ρ), ind D/ = − dim(V ), if Alt(µ) = Alt(λ + ρ), (9.1.17) s µ  λ  0, if Alt(µ) = 0, where Vλ is the irreducible representation space of G with highest weight λ. As a simple example, let G = SU(2). Then T is isomorphic to U(1); and we have ΛT ' Z. In fact, we did some explicit calculations in Section 3.2 and saw that

ΛT = Z · ρ. Let us choose the nonnegative integral multiples of ρ as the dominant weights. Let Vn denote the irreducible representation of G with highest weight nρ, n > 0. Equation 9.1.17, in this case, simpliﬁes to

− dim(V(n−1)ρ), if n > 0, ind D/ = 0, if n = 0, s nρ   dim(V−(n+1)ρ), if n < 0. By Equation 3.2.12,  dim(Vnρ) = n + 1. Thus, IndT D/ + = (−n)[Cnρ]. n∈ XZ Let H be the vector in t such that ρ(H) = 1. Then, the character of the representation of T on Cnρ is given by inx χn(exp(xH)) = e . In terms of these characters,

IndT D/ + = − nχn. n∈ XZ This is just the derivative of the distribution i χ ; but χ n∈Z n n∈Z n is, by Fourier theory, the Dirac delta distribution δe supported at the identity. Hence, P P 0 IndT D/ + = iδe.

142 9.2 THEDISTRIBUTIONALINDEXOF D/ (g, k)

9.2.1 We are interested in the case where K is a closed connected subgroup of a compact connected Lie group G and the K-manifold is M := G, on which K is acting by right-multiplication. We assume that K is of maximal rank and that G/K is a spin manifold. We choose a bi-invariant metric on G such that the volume form it generates for G and K assign unit volume to them. We continue to use the notations set up in Section 7.2.1. But for simplicity, we assume that the Cl(p)-module E is just the spinor Ad space S. Then K acts on S by ν := Ad,f which is the lift of K − SO(p) to Spin(p). 2 Consider the trivial bundle G × S G. Denote by Γ (G→× S) the 2 2 L -closure of Γ(G × S). We let K act on Γ (G × S) as follows: For k 2 in K and σ in Γ (G × S), → k · σ : g 7 ν(k)σ(gk). Following the notation in Section 9.1.10, we denote this representation by → 2 R : K Aut(Γ (G × S)).

For f in C (K), the operator Rf deﬁned by Equation 9.1.11 acts 2 → on Γ (G × S) ∞by

Rfσ : g 7 f(k)ν(k)σ(gk) dk. (9.2.2) ZK The differential operator→ we have in mind is the relative Dirac K operator D/ (g, k) on the space Γ(G×S) of K-invariant sections of G× S; this operator is, a priori, only transversally elliptic (Lemma 9.2.4). But, as we have seen in Proposition 6.3.14, D/ (g, k) is equivalent to the Dirac operator D/ g/k on Γ(G ×ν S), so D/ (g, k) is effectively elliptic. 2 K Moreover, D/ (g, k) admits a unique self-adjoint extension Γ (G × S) (see Section 8.1.1); we shall denote this extension by D/ . Because p is even-dimensional, the spinor space is Z/2Z-graded (see Section 5.1.44): + − S = S ⊕ S . This induces a grading on the space of sections: 2 K 2 + K 2 − K Γ (G × S) = Γ (G × S ) ⊕ Γ (G × S ) . The Dirac operator D/ is an odd operator (see Section 6.2.6). We write 0 D/ D/ = − . D/ + 0

The restriction D/ + of D/ to the even subspace is also transversally elliptic. The distributional index of D/ + is

IndK D/ + = [ker D/ +] − [ker D/ −].

143 9.2.3 DEFINITION. We deﬁne the distributional index of D/ as

IndK D/ := IndK D/ + = [ker D/ +] − [ker D/ −].

9.2.4 LEMMA. The Dirac operator D/ is transversally elliptic.

Proof. Since D/ is invariant under left-translations on G, it is sufficient to check its transversal ellipticity at the identity of G. With respect to the orthogonal decomposition g = k ⊕ p, the cotangent space of G ∗ ∗ ∗ ∗ at the identity decomposes into g = k ⊕ p . Then the fiber of TKG ∗ dim p over the identity can be naturally identified with p . Let { Yi }i=1 be an orthonormal basis for p. Then, as we have seen in the proof of Proposition 6.3.14, D/ at the identity can be expressed as dim p dim p 1 D/ = c(Y )∂ + c(Y γp(Y )), i Yi 3 i i i=1 i=1 X X where c is the Clifford action and ∂Yi denotes the directional derivative with respect to Yi. From this expression it is clear that D/ is transversally elliptic.

9.2.5 NOTATION. Let ∆G be the Laplacian on G. We set

1 1 2 2 L := ∆ + kρ k − kρ k . 2 G 2 k g 2 By Equation 7.2.5, L is equal to D/ on the domain of D/ .

9.2.6 LEMMA. Let f be a smooth function on K. Let Rf be the operator 2 K on Γ (G × S) deﬁned by Equation 9.2.2. Then tL hIndK D/ , fi = Str(Rfe ), 2 K where the super-trace on the right-hand side is over Γ (G × S) . Proof. By Equation 9.1.12, we have + − hIndK D/ , fi = tr(Rf ) − tr(Rf ), (9.2.7) We wish to show that + − tL tr(Rf ) − tr(Rf ) = Str(Rfe ). (9.2.8)

: 2 K First, note that Rf acts as a scalar C = K f(k) dk on Γ (G × K) . So tr(R+) − tr(R−) is equal to C times the (graded) index of L. Hence, f f R + − tL tL tr(Rf ) − tr(Rf ) = C inds L = C Str(e ) = Str(Rfe ).

9.2.9 PROPOSITION. Let f be a smooth function on K. Let rt be the heat convolution kernel of L. Then

−1 hIndK D/ , fi = Str rt(k)f(k)ν(k) dk dg¯ ZG/K ZK where g¯ := gK.

144 2 K : Proof. Recall that Rf acts on Γ (G × S) as the scalar C = K f(k) dk. So tL tL R Str(Rfe ) = C Str(e ) = C trs(Rt(g, g)) dg, ZG where Rt is the heat kernel of L. By Equation 7.1.4, trs(Rt(g, g)) = trs(Rt(e, e)). Since rt(x) = Rt(e, x) by deﬁnition, trs(Rt(g, g)) = trs(rt(e)). Hence,

tL Str(Rfe ) = C trs(rt(e)) dg = C trs(rt(e)) dk dg.¯ ZG ZG/K ZK Because the measure on K is normalized,

tL Str(Rfe ) = C trs(rt(e)) dg.¯ ZG/K

Since C = K f(k) dk,

R tL Str(Rfe ) = f(k) trs(rt(e)) dk dg.¯ ZG/K ZK −1 Owing to Lemma 8.2.15, rt(e) = rt(k)ν(k) ; hence,

tL −1 Str(Rfe ) = f(k) trs(rt(k)ν(k) ) dk dg.¯ ZG/K ZK

9.2.10 Since K acts freely on G, the distributional index IndK D/ is supported at the identity of K (Corollary 9.1.15). Recall the distribution theoretic description of the Duﬂo isomorphism for the Lie group K (see Section 4.1.22): 0 ∼ 0 Duf = exp∗ ◦ jk : E0(k) − Ee(K). (9.2.11)

Here exp∗ is the pushforward map of distributions along the exponential map, jk is the multiplication by the→ function deﬁned by the 0 power series 4.1.24, E0(k) is the algebra of distributions on k with 0 support {0}, and Ee(K) is the algebra of distributions on K with sup- 0 port {e}. Since IndK D/ is in Ee(K), we can apply the inverse of the −1 0 Duﬂo isomorphism to it and obtain Duf IndK D/ in E0(k). −1 Let us examine the pairing of Duf IndK D/ with a test function. To that end, let V be a neighborhood of 0 in k that is mapped diffeomorphically by the exponential map onto a neighborhood U of e in K; denote this local diffeomorphism by ∼ expV : V − U.

: −1 Let log = expV . We have the induced pushforward map of distributions, → 0 0 log∗ : D (U) D (V).

→

145 This makes the following diagram commutative:

log D0(V) o ∗ D0(U) O ∼ O ? exp ? E0 (k) ∗ / E0 (K) 0 ∼ e 0 The horizontal maps are linear isomorphisms. Thus, for δ in Ee(K), we have (exp∗ ◦ log∗)(δ) = δ. So −1 −1 (Duf ◦ jk ◦ log∗)(δ) = (exp∗ ◦ jk ◦ jk ◦ log∗)(δ) = δ. (9.2.12) Hence, −1 −1 (jk ◦ log∗)(δ) = Duf (δ). (9.2.13) Now let φ be a smooth function on k. Let ψ be any bump function supported on V such that ψ(X) = 1 for X near 0 in k. Then, since −1 Duf IndK D/ is supported at 0, −1 −1 hDuf IndK D/ , φi = hDuf IndK D/ , φψi. Then, by Equation 9.2.13, −1 −1 ∗ −1 hDuf IndK D/ , φi = hjk log∗ IndK D/ , φψi = hIndK D/ , log (jk φψ)i. Finally, by Proposition 9.2.9,

−1 ∗ −1 −1 hDuf IndK D/ , φi = trs rt(k) log (jk φψ)(k)ν(k) dk dg.¯ ZG/K ZK (9.2.14) If φ is a polynomial on k that is invariant under ad(k)-action, then it is subject to the Chern-Weil homomorphism 6.1.27. Consider the integral Chern-Weil homomorphism (this is not a standard terminol- ogy; see the remark on page 99 for the reason we call it this way) S(k∗) H∗(G/K), (9.2.15) φ 7 φ,ˆ → obtained by putting an extra factor of (2πi)−1 in front of the curvature 2-form Θ in Equation 6→.1.28. Thus, if φ is a homogeneous polynomial of degree n, the differential form φˆ at the identity coset e¯ satisfies −n φˆ e¯ = (2πi) A(φ), (9.2.16) where A is the algebra homomorphism defined in Definition 8.2.36. (Here we are identifying the cotangent space of G/K ate ¯ with p∗, where the p is the orthogonal complement of k in g.) We then have the following result (This theorem was suggested by N. Higson):

9.2.17 THEOREM. Let φ be an invariant polynomial on k. Let φˆ be the characteristic class of the principal K-bundle G G/K obtained by the image of φ under the integral Chern-Weil homomorphism 9.2.15. Let →

146 Aˆ be the Hirzebruch Aˆ -class of the tangent bundle of G/K. Let IndK D/ K be the distributional index of the relative Dirac operator on Γ(G × S) . The pairing of φ with the inverse image of IndK D/ under the Duﬂo isomorphism 9.2.11 satisﬁes −1 ˆ ˆ hDuf IndK D/ , φi = hA ` φ, [G/K]i. (9.2.18) Here Aˆ `φˆ is the cup product of Aˆ and φˆ, and [G/K] is the fundamental homology class of G/K associated to the quotient measure on G/K with respect to the normalized Haar measures on G and K.

Proof. By Equation 9.2.14,

−1 ∗ −1 −1 hDuf IndK D/ , φi = trs rt(k) log (jk φψ)(k)ν(k) dk dg.¯ ZG/K ZK (9.2.19) Let us write the integral over K as

∗ : −1 −1 I(t) = trs rt(k) log (jk φψ)(k)ν(k) dk. ZK We claim that, for t 0+, top I(t) vol = Aˆ φˆ + O(t), (9.2.20) → where vol is the volume form of G/K associated with the quotient top measure on G/K, and Aˆ φˆ is the top degree part of the exterior product of Aˆ and φˆ. Assume, for the moment, that the claim is true; then the theorem follows, since the left-hand side of Equation 9.2.19 is independent of t. In order to prove our claim, let us change the domain of the integral I(t), from K to k. We have

−γp(X) I(t) = trs rt(exp X)jk(X)φ(X)ψ(X)e dX. (9.2.21) Zk For t 0+, the convolution kernel rt is of O(t ) outside any neighborhood of the identity; and since ψ(X) = 1 for∞X near 0, we have → −γp(X) I(t) ∼ trs rt(exp X)jk(X)φ(X)e dX. (9.2.22) Zk By Theorem 7.2.15 and Equation 8.2.22, we have

2 p tkρkk /2 g −1 −γ (X) I(t) ∼ e ht (X)jg/k(X) φ(X) trs(e ) dX. Zk We may replace γp with λp because the super-trace only cares about the top degree part;

2 p tkρkk /2 g −1 −λ (X) I(t) ∼ e ht (X)jg/k(X) φ(X) trs(e ) dX. Zk Apply Lemmas 8.2.25 and 8.2.35 exactly as done in the proof of The- orem 8.2.40; then, we get I(t) = (2πi)− dim(G/K) (j−1 ) (φ) top + O(t). vole¯ A g/k A e¯

147 By Equations 8.2.39 and 9.2.16, we have I(t) = Aˆ φˆ top + O(t). vole¯ e¯ (9.2.23) This is Equation 9.2.20 ate ¯. By homogeneity, it must hold every- where on G/K. The theorem now follows.

148 BIBLIOGRAPHY

[1] I. Agricola, Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. Math. Phys. 232 (2003), no. 3, 535–563. MR1952476 (2004c:53066) [2] A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, In- vent. Math. 139 (2000), no. 1, 135–172, DOI 10.1007/s002229900025. MR1728878 (2001j:17022) [3] , Lie theory and the Chern-Weil homomorphism, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 303–338, DOI 10.1016/j.ansens.2004.11.004 (En- glish, with English and French summaries). MR2144989 (2006d:53020) [4] , On the Kashiwara-Vergne conjecture, Invent. Math. 164 (2006), no. 3, 615–634, DOI 10.1007/s00222-005-0486-4. MR2221133 (2007g:17017) [5] M. F. Atiyah, The index of elliptic operators, Fields Medallists’ lectures, World Sci. Ser. 20th Century Math., vol. 5, World Sci. Publ., River Edge, NJ, 1997, pp. 115–127; reprinted in Collected works. Vol. 3, Oxford Science Publi- cations, The Clarendon Press Oxford University Press, New York, 1988. MR1622942 [6] M. F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathe- matics, Vol. 401, Springer-Verlag, Berlin, 1974. MR0482866 (58 #2910) [7] M. F. Atiyah and R. Bott, A Lefschetz ﬁxed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR0232406 (38 #731) [8] M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. MR0650828 (58 #31287) [9] M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR0463358 (57 #3310) [10] M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433. MR0157392 (28 #626) [11] C. Bär, The Dirac operator on homogeneous spaces and its spectrum on 3- dimensional lens spaces, Arch. Math. (Basel) 59 (1992), no. 1, 65–79, DOI 10.1007/BF01199016. MR1166019 (93h:58156) [12] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004. Corrected reprint of the 1992 original. MR2273508 (2007m:58033) [13] N. Berline and M. Vergne, A computation of the equivariant index of the Dirac operator, Bull. Soc. Math. France 113 (1985), no. 3, 305–345 (English, with French summary). MR834043 (87f:58146) [14] G. Birkhoff, Representability of Lie algebras and Lie groups by matrices, Ann. of Math. (2) 38 (1937), no. 2, 526–532, DOI 10.2307/1968569. MR1503351 [15] O. Bonnet, Mémoire sur la théorie générale des surfaces, J. Ecole Polytechnique 19 (1848), no. 32, 1–146. [16] R. Bott, The index theorem for homogeneous differential operators, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 167–186. MR0182022 (31 #6246)

149 [17] N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Math- ematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR1890629 (2003a:17001) [18] D. Bump, Lie groups, Graduate Texts in Mathematics, vol. 225, Springer- Verlag, New York, 2004. MR2062813 (2005f:22001) [19] E. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France 41 (1913), 53–96 (French). MR1504700 [20] H. Cartan, Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie (espaces fi- brés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 15–27 (French). MR0042426 (13,107e) [21] , La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 57–71 (French). MR0042427 (13,107f) [22] J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam, 1975. North-Holland Mathemat- ical Library, Vol. 9. MR0458335 (56 #16538) [23] S.-s. Chern, On the curvatura integra in a Riemannian manifold, Ann. of Math. (2) 46 (1945), 674–684. MR0014760 (7,328c) [24] , Characteristic classes of Hermitian manifolds, Ann. of Math. (2) 47 (1946), 85–121. MR0015793 (7,470b) [25] C. Chevalley, The algebraic theory of spinors, Columbia University Press, New York, 1954. MR0060497 (15,678d) [26] , The algebraic theory of spinors and Clifford algebras, Springer-Verlag, Berlin, 1997. Collected works. Vol. 2; Edited and with a foreword by Pierre Cartier and Catherine Chevalley; With a postface by J.-P. Bourguignon. MR1636473 (99f:01028) [27] S. C. Coutinho, A primer of algebraic D-modules, London Mathematical So- ciety Student Texts, vol. 33, Cambridge University Press, Cambridge, 1995. MR1356713 (96j:32011) [28] J. Dieudonné, Treatise on analysis. Vol. III, Academic Press, New York, 1972. Translated from the French by I. G. MacDonald; Pure and Applied Mathe- matics, Vol. 10-III. MR0350769 (50 #3261) [29] P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. A 117 (1928), no. 778, 610–624. [30] , The Principles of Quantum Mechanics, Oxford, at the Clarendon Press, 1958. 4th ed (Revised). [31] D. S. Dummit and R. M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons Inc., Hoboken, NJ, 2004. MR2286236 (2007h:00003) [32] M. Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 265–288 (French, with English summary). MR0444841 (56 #3188) [33] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR0405514 (53 #9307) [34] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR1738431 (2001j:22008) [35] J. J. Duistermaat, The heat kernel Lefschetz fixed point formula for the spin-c Dirac operator, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011. Reprint of the 1996 edition. MR2809491 (2012b:58032) [36] L. Euler, Methodus universalis serierum convergentium summas quam prox- ime inveniendi, Commentarii academie scientiarum Petropolitanae 8 (1736), 147–158; reprinted in Opera Omnia. Vol. XIV.

150 [37] , Methodus universalis series summandi ulterius promota, Commen- tarii academie scientiarum Petropolitanae 8 (1736), 147–158; reprinted in Opera Omnia. Vol. XIV. [38] H. D. Fegan, The heat equation and modular forms, J. Differential Geom. 13 (1978), no. 4, 589–602 (1979). MR570220 (81k:22006) [39] , The fundamental solution of the heat equation on a compact Lie group, J. Differential Geom. 18 (1983), no. 4, 659–668 (1984). MR730921 (85j:58140) [40] M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), no. 1, 106–146, DOI 10.1016/0022-1236(78)90058-7. MR513481 (80f:43022) [41] D. S. Freed, Flag manifolds and infinite-dimensional Kähler geometry, Infinite- dimensional groups with applications (Berkeley, Calif., 1984), Math. Sci. Res. Inst. Publ., vol. 4, Springer, New York, 1985, pp. 83–124. MR823316 (87k:58020) [42] H. Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathe- matics, Vol. 35, Academic Press, New York, 1969. MR0260926 (41 #5546) [43] T. Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. MR1777332 (2001c:58017) [44] J. Fröhlich, O. Grandjean, and A. Recknagel, Supersymmetric quantum theory and non-commutative geometry, Comm. Math. Phys. 203 (1999), no. 1, 119– 184, DOI 10.1007/s002200050608. MR1695097 (2001d:58025) [45] C. F. Gauss, Disquisitiones generales circa superficies curvas, Comment. Soc. Gottingen. (classis mathematicae) 6 (1828), 99-146. [46] L. Gårding, Dirichlet’s problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55–72. MR0064979 (16,366a) [47] E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah- Singer index theorem, Comm. Math. Phys. 92 (1983), no. 2, 163–178. MR728863 (86a:58104) [48] P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math. 10 (1973), 344–382. MR0324731 (48 #3081) [49] C. Goldstein and G. Skandalis, An interview with Alain Connes, Newsletter of EMS. 63 (2007), 25–30. [50] Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28–96. MR0044515 (13,428c) [51] , Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117–201. MR0439994 (55 #12875) [52] S. Helgason, Differential operators on homogenous spaces, Acta Math. 102 (1959), 239–299. MR0117681 (22 #8457) [53] S. Helgason, Analysis on Lie groups and homogeneous spaces, American Mathematical Society, Providence, R.I., 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 14. MR0316632 (47 #5179) [54] , Differential geometry, Lie groups, and symmetric spaces, Pure and Ap- plied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. MR514561 (80k:53081) [55] , Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR754767 (86c:22017)

151 [56] N. Higson, The local index formula in noncommutative geometry, Contem- porary developments in algebraic K-theory, ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 443–536 (electronic). MR2175637 (2006m:58039) [57] F. Hirzebruch, Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 110–114. MR0074086 (17,535a) [58] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9, Springer- Verlag, Berlin, 1956 (German). MR0082174 (18,509b) [59] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad- uate Texts in Mathematics, vol. 9, Springer-Verlag, New York, 1978. Second printing, revised. MR499562 (81b:17007) [60] M. Kashiwara and M. Vergne, The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272. MR0492078 (58 #11232) [61] A. W. Knapp, Advanced real analysis, Cornerstones, Birkhäuser Boston Inc., Boston, MA, 2005. Along with a companion volume Basic real analysis. MR2155260 (2006c:26001) [62] , Lie groups beyond an introduction, 2nd ed., Progress in Mathe- matics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002. MR1920389 (2003c:22001) [63] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Wi- ley Classics Library, John Wiley & Sons Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR1393940 (97c:53001a) [64] B. Kostant, Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the ρ-decomposition C(g) = End Vρ ⊗ C(P), and the g-module structure of V g, Adv. Math. 125 (1997), no. 2, 275–350, DOI 10.1006/aima.1997.1608. MR1434113 (98k:17009) [65] , A cubic Dirac operator and the emergence of Euler number multi- plets of representations for equal rank subgroups, Duke Math. J. 100 (1999), no. 3, 447–501, DOI 10.1215/S0012-7094-99-10016-0. MR1719734 (2001k:22032) [66] B. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology, and inﬁnite-dimensional Clifford algebras, Ann. Physics 176 (1987), no. 1, 49– 113, DOI 10.1016/0003-4916(87)90178-3. MR893479 (88m:58057) [67] V. Lafforgue, Banach KK-theory and the Baum-Connes conjecture, (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 795–812. MR1957086 (2003k:19006) [68] G. D. Landweber, Dirac operators on loop spaces, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–Harvard University. MR2699363 [69] H. B. Lawson Jr. and M.-L. Michelsohn, Spin geometry, Princeton Math- ematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR1031992 (91g:53001) [70] S. Lefschetz, Intersections and transformations of complexes and manifolds, Trans. Amer. Math. Soc. 28 (1926), no. 1, 1–49, DOI 10.2307/1989171. MR1501331 [71] H. A. Lorentz, Alte und neue Fragen der Physik, Physikal. Zeitschr. 11 (1910), 1234–1257. [72] I. G. Macdonald, The volume of a compact Lie group, Invent. Math. 56 (1980), no. 2, 93–95, DOI 10.1007/BF01392542. MR558859 (81h:22018) [73] Colin Maclaurin, A Treatise of Fluxions, Edinburgh, 1742.

152 [74] H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Lapla- cian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR0217739 (36 #828) [75] S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian J. Math. 1 (1949), 242–256. MR0031145 (11,108b) [76] S. Morita, Geometry of differential forms, Translations of Mathematical Mono- graphs, vol. 201, American Mathematical Society, Providence, RI, 2001. Translated from the two-volume Japanese original (1997, 1998) by Teruko Nagase and Katsumi Nomizu; Iwanami Series in Modern Mathematics. MR1851352 (2002k:58001) [77] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR0318398 (47 #6945) [78] V. K. Patodi, An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds, J. Differential Geometry 5 (1971), 251–283. MR0290318 (44 #7502) [79] F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe, Math. Ann. 97 (1927), no. 1, 737– 755, DOI 10.1007/BF01447892 (German). MR1512386 [80] H. Poincaré, Sur les groupes continus, Trans. Cambr. Philos. Soc. 18 (1900), 220–225. [81] A. V. Pukhlikov and A. G. Khovanski˘ı, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz 4 (1992), no. 4, 188–216 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 4, 789–812. MR1190788 (94c:14044) [82] B. Riemann, Theorie der Abel’schen Functionen, J. reine angew. Math. 54 (1857), 115–155. [83] A. Robert, Introduction to the representation theory of compact and locally compact groups, London Mathematical Society Lecture Note Series, vol. 80, Cambridge University Press, Cambridge, 1983. MR690955 (84h:22012) [84] G. Roch, Ueber die Anzahl der willkürlichen Constanten in algebraischen Func- tionen, J. reine angew. Math. 64 (1865), 372–376. [85] J. Roe, Elliptic operators, topology and asymptotic methods, 2nd ed., Pitman Research Notes in Mathematics Series, vol. 395, Longman, Harlow, 1998. MR1670907 (99m:58182) [86] W. Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. MR1157815 (92k:46001) [87] R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR0237943 (38 #6220) [88] J.-P. Serre, Complex semisimple Lie algebras, Springer-Verlag, New York, 1987. Translated from the French by G. A. Jones. MR914496 (89b:17001) [89] M. E. Taylor, Partial differential equations. I, Applied Mathematical Sci- ences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR1395148 (98b:35002b) [90] F. Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, Plenum Press, New York, 1980. Pseudodifferential operators; The Univer- sity Series in Mathematics. MR597144 (82i:35173) [91] L. W. Tu, An introduction to manifolds, 2nd ed., Universitext, Springer, New York, 2011. MR2723362 (2011g:58001) [92] F. W. Warner, Foundations of differentiable manifolds and Lie groups, Grad- uate Texts in Mathematics, vol. 94, Springer-Verlag, New York, 1983. Cor- rected reprint of the 1971 edition. MR722297 (84k:58001)

153 [93] A. Weil, Géométrie différentielle des espaces ﬁbrés, Letters to Chevalley and Koszul (1949); reprinted in Œuvres scientiﬁques. Collected papers. Volume I (1926–1951), Springer-Verlag, Berlin, 2009. pp. 422–436. [94] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer par- tieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479, DOI 10.1007/BF01456804 (German). MR1511670 [95] , Über die Abhängigkeit der Eigenschwingungen einer Membran und deren Begrenzung, J. Reine Angew. Math. 141 (1912), 1–11, DOI 10.1515/CRLL.1912.141.1 (German). [96] H. Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I, Math. Z. 23 (1925), no. 1, 271–309, DOI 10.1007/BF01506234 (German). MR1544744 [97] , Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. II, Math. Z. 24 (1926), no. 1, 328–376, DOI 10.1007/BF01216788 (German). MR1544769 [98] , Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. III, Math. Z. 24 (1926), no. 1, 377–395, DOI 10.1007/BF01216789 (German). MR1544770 [99] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939. MR0000255 (1,42c) [100] E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 117 (1937), 152–160.

[101] S. Zelditch, Macdonald’s identities and the large N limit of YM2 on the cylinder, Comm. Math. Phys. 245 (2004), no. 3, 611–626, DOI 10.1007/s00220-003- 1027-x. MR2045685 (2005g:81198)

154 VITA

SEUNGHUNHONG

Seunghun Hong’s curiosity in science was aroused by the visit of Halley’s comet in 1986; it eventually led him to enter Seoul National University as a physics major. While yet a university student, he ful- ﬁlled his obligatory military service at the 121st General Hospital as a radiographic technician. He received his Bachelor of Science in 2002, and entered Tufts University for graduate studies in physics. While taking some graduate courses in mathematics, in particular, a course on abstract algebra taught by Loring W. Tu, he got captivated by the beauty of mathematics and decided to switch his discipline. After receiving his Master of Science in physics in 2004, he began his formal graduate studies in mathematics. He wrote a thesis under the guidance of Loring W. Tu and received a Master of Arts in mathematics in 2006. Intrigued by the work of Alain Connes, he decided to pursue his doctoral studies in the area of noncommutative geometry. He entered the Graduate School at PENN STATE, and became a student of Nigel Higson. Awards won by him include Honorable Mention in the 20th Na- tionwide University Students Contest of Mathematics held by the Ko- rean Mathematical Society, The Korean Honor Scholarship from the Ambassador of Korea in the U.S.A., and the Charles H. Hoover Memo- rial Award from the Department of Mathematics at PENN STATE. He has recently accepted a postdoctoral position in the Mathe- matical Institute at the University of Göttingen in Germany.